As I'm sure most of you all were aware, today, March 14th or 3/14 (as we say in the USA) is Pi Day! At 1:59:26 AM and PM, the date and time conspired to produce the first 8 digits of the decimal expansion of Pi, the ratio of the circumference of any circle to the diameter of that circle! I trust you all celebrated accordingly!
On the appendix cancer front, we (Caroline, Burr and Cynthia, and Brother Norman, and me) met with my surgical oncologist, Dr. Paul Mansfield, and he recommended surgery to remove all the tumors from my abdominal cavity. The surgery is tentatively set for Tuesday, April 12, 2011. Depending on how involved the lymph system is, Dr. Mansfield will make the call during the surgery about whether or not to use the Hyperthermal IntraPeritoneal Chemotherapy (HIPEC).
Great news! Thank you, Lord, for all your many blessings, most particularly for Caroline and the rest of my loving family.
In case you were wondering, especially on PI Day, here are some more decimal digits of PI:
3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 ...
Randy and Caroline
A lovely July in Seattle!
Monday, March 14, 2011
Sunday, March 6, 2011
Great News About My CT Scan!
I finally got my long-awaited CT scan last night around 8:00PM (it was originally scheduled for 6:00PM, but that's par for the course!--it's why we're called patients, I guess!) and, fortunately, the pros at MDACC were able to quickly analyze the CT scan results overnight so that when we met with my wonderful colorectal oncologist, Dr. Imad Shureiqi, today at 1:30PM he was able to give us (me, Caroline, Burr, and Cynthia) the good news--the tumors had not grown at all since the previous CT scan taken on December 13, 2010, there were no new metastatic nodules, and the tumors had not spread beyond the peritoneum!!! There were no "spots" on my liver or pancreas or kidneys or lungs or heart or anywhere outside my abdominal cavity! The primary appendiceal adenocarcinoma was about the same size as it was in December, maybe even slightly smaller (I can't recall the exact dimensions, but I don't think any of the lineal measures of the appendix tumor were more than 3 centimeters, a little over an inch, about 1.18 inches at 2.54 centimeters to an inch, so the primary tumor has never been more than 27 cubic centimeters in volume, or 1.64 cubic inches), and the 4 metastatic nodules on my peritoneum were all still each less than about 1 cubic centimeter in size, which is about 0.061 cubic inches. And there aren't any more nodules! No new nodules! The lymph nodes around the appendix are no larger than they had been before! The chemotherapy, FOLFOX6, has apparently succeeded in stopping the cancer in its tracks! Praise God! Hallelujah! Hosanna in the Highest! Thank you Lord!
The next step is to meet again with my surgical oncologist, Dr. Paul Mansfield, to discuss whether it makes sense to pursue surgical options, such as Hyperthermal Intraperitoneal Chemotherapy (HIPEC), at this time. Hopefully, Dr. Mansfield will want to do the surgery! It's not any easy surgery to endure. It can take up to 14 hours! That's a long time to be under general anesthesia! The HIPEC part of the surgery itself includes 90 minutes or so of my being gently rocked back and forth while the warm chemo bath is sloshing around inside my abdominal cavity! The direct application of the chemo to the affected tissues is much more effective than chemo delivered intravenously, which, because it's necessarily delivered systemically throughout the whole body cannot be nearly as concentrated or as strong, because of the deliterious consequences of the deadly chemo cocktail. Supposedly, one-third of the patients who under go the full HIPEC treatment experience a complete remission of their cancers with no reoccurence of the cancers! I intend to be in that one-third! Those are good odds, indeed! Much better than the odds of winning the lottery, for example!
Needless to say, we're all very pleased by all this! Ecstatic, you might say! Euphoric! I'm still at MDACC getting my fifth FOLFOX6 chemo treatment, fifth out of twelve altogether. Thanks to Dave and Jo and Brian and Celeste, I'm able to post this in real time using our beloved Paddy the iPad! We love Paddy and can't imagine what it would be like living without it!
On an entirely unrelated note, for all of you who are wondering about the fate and robustness of Furlong's Conjecture (see previous posting of the same name--basically, I modestly conjectured that there will always be at least two twin prime pairs between the squares of successive prime numbers!), here's the latest and greatest that I've been able to coax out of Excel (equipped with an add-on that permits virtually unlimited precision integer operations and many very handy number theory operations to be performed--for more, see Joe Crump's Immortal Theory Number Theory website), although not without a lot of tsuris (Yiddish for trouble) and much kutput (Gujarati for trouble):
My nifty Factorizer program (not programmed by me, of course, but able to be purchased for a modest sum at Factorizer) found 531 prime pairs in the range from 2,141,005,441=46,271^2 [2.141005411 billion or 2.141005411x10^9, a 10 digit number] through 2,141,190,529=46,273^2 [2.141190529 billion or 2.141190529x10^9, a 10 digit number]. Needless to say, my conjecture is certainly robust enough (with 529=23^2 prime pairs to spare!) out to 46,271^2 and 46,273^2!!! Unfortunately, Factorizer does not do numbers larger than the Mersenne prime M_31=2^31-1=2,147,483,647 so I was forced to turn to my makeshift Excel number theory kluge!
Between (2,141,190,431)^2=4,584,696,461,805,965,761 [4.584696461805965761 quintillion or 4.584696461805965761x10^18, a 19 digit number] and
(2,141,190,433)^2=4,584,696,470,370,727,489 [4.584696470370727489 quintillion or 4.584696470370727489x10^18, a 19 digit number] there are at least 1863 twin primes between (2,141,190,431)^2=4,584,696,461,805,965,761 and 4,584,696,461,808,572,881 (with 8,564,761,728 - 2,607,120 = 8,562,154,608 [8.562154608 billion or 8.562154608x10^9, a 10 digit number] numbers in the range to spare)!!! The range of numbers chosen is not completely arbirtary! It merely reflects the fact that Excel only permits 65,536=2^16 rows in any worksheet and I'm too lazy to start another column of prime numbers, so for each range of numbers I'm really only sampling the first 60,000 or so numbers in each range (for reasons too unimportant to go into it just so happens that I start each column of my primes between squares of twin primes at row 5,536 or thereabouts, so I don't even get the full 65,536 numbers that I could otherwise get in each of columns)!
Between (4,584,696,461,808,572,489)^2=21,019,441,646,920,043,379,214,312,327,967,655,121
[21.019441646920043379214312327967655121 undecillion or 2.1019441646920043379214312327967655121x10^37, a 38 digit number] and (4,584,696,461,808,572,491)^2 =21,019,441,646,920,043,397,553,098,175,201,945,081 [21.019441646920043397553098175201945081 undecillion or 2.1019441646920043397553098175201945081x10^37, a 38 digit number] there are at least 940 twin primes between (4,584,696,461,808,572,489)^2 =21,019,441,646,920,043,379,214,312,327,967,655,121 and
21,019,441,646,920,043,379,214,312,327,972,886,863 (with 18,338,785,847,234,289,960 -
5,231,742 = 18,338,785,847,229,058,218 [18.338785847229058218 quintillion or 1.8338785847229058218x10^19, a 20 digit number] in the range to spare)!!!
At this point, a little explanation of terminology may be in order! The issue is how to express large numbers in English! John Conway and Richard Guy have a very good explanation of how we should go about doing this for numbers as large as you like in their superb book The Book of Numbers. Among other things, Conway and Guy point out that what it really all boils down to is coming up with reasonable names for the Nth zillion numbers that are powers of 10 that go like 10^(3N+3), according to the American naming system and not the British naming system, which uses 10^6N instead! The first few of these are well known and beloved by all: one million (1,000,000=10^6, a 7 digit number) for N=1, one billion (1,000,000,000=10^9, a 10 digit number) for N=2, one trillion (1,000,000,000,000=10^12, a 13 digit number) for N=3, one quadrillion (1,000,000,000,000,000=10^15, a 16 digit number) for N=4, one quintillion (1,000,000,000,000,000,000=10^18, a 19 digit number) for N=5, one sextillion (1,000,000,000,000,000,000,000=10^21, a 22 digit number) for N=6, one septillion (1,000,000,000,000,000,000,000,000=10^24, a 25 digit number) for N=7, one octillion (1,000,000,000,000,000,000,000,000,000=10^27, a 28 digit number) for N=8, one nonillion (1,000,000,000,000,000,000,000,000,000,000=10^30, a 31 digit number) for N=9, and one decillion (1,000,000,000,000,000,000,000,000,000,000,000=10^33, a 34 digit number) for N=10. Notice that the number of triple zero clusters is always one more than the value of N! That's because of the "+3" in the (3N+3) exponent in 10^(3N+3)!
When would you ever have a chance to use such names in everyday conversation? Well, the federal budget is in the trillions (a few times 10^12), for example. The mass of the Earth's atmosphere is roughly 4 quintillion (4x10^18) kilograms. The moon's mass is about 70 sextillion (7x10^22) kilograms. Avogadro's number is 602.214179(30) sextillion (6.02214179(30)x10^23), which is the number of carbon-12 atoms in 12 grams of pure carbon-12, for example (the "(30)" at the end of the number 6.02214179 reflects a "standard experimental uncertainty" of +/-30 in the last two digits [it understood that the number "(30)" in parentheses is the numerical value of the standard uncertainty, the estimated standard deviation, where the N data points are assumed to be adequately characterized by a Gaussian or normal probability distribution that looks like a Bell curve when plotted out, centered on the average or mean value X_average=Sum from i=1 to i=N of (X_i)/N, with a spread about the average value given by s=the square root of the Sum from i=1 to i=N of {(X_i-X_average)^2}/(N-1), the variance normalized by one less than the number N of data points {for exceedingly obscure reasons related to something called the "Student's t-distribution" that I won't go into, which is a welcome relief to all concerned!}] referred to the corresponding last digits of the quoted result--this implies that it is believed with an approximate level of confidence of 68.27% that the "true" value of Avogadro's number probably lies between 6.012214149x10^23 and 6.012214209x10^23, that it is believed with an approximate level of confidence of 95% that the "true" value of Avogadro's number probably lies between 6.012214119x10^23 and 6.012214239x10^23 [from an "expanded" experimental uncertainty of about +/-60 in the last two digits that may readily obtained by multiplying the quoted value of the "standard experimental uncertainty" by 1.960, which I've conveniently rounded up to 2.000, keeping 3 significant digits in the rounding and technically boosting my level of confidence from 95% to 95.45%, as may be confirmed here, at page 82, Table G.1], and that it is believed with an approximate level of confidence of 99% that the "true" value of Avogadro's number probably lies between 6.012214089x10^23 and 6.012214269x10^23 [from an "expanded" experimental uncertainty of about +/-90 in the last two digits that may readily obtained by multiplying the quoted value of the "standard experimental uncertainty" by 2.576, which I've conveniently rounded up to 3.000, keeping 3 significant digits in the rounding and technically boosting my level of confidence from 99% to 99.73%]). As pointed out in the marveously useful book Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin, if you had an Avogadro's number of fairly fat cats (at 10 kilograms each!), they would have a mass of just about 6 septillion (6x10^24) kilograms, which happens to be the mass of the entire Earth! The mass of Jupiter is in the ballpark of one octillion (10^27) kilograms, while the mass of our Sun is approximately 2 nonillion (2x10^30) kilograms or 2 decillion (2x10^33) grams! There are in the neighborhood of 100 billion (10^11) stars in the Milky Way galaxy so it has a mass of 200 duodecillion (2x10^41) kilograms or 200 tredecillion (2x10^44) grams. There are on the order of 100 billion (10^11) galaxies in the whole known Universe, as far as we know, so that the whole known Universe should have a mass of about 20 sedecillion (2x10^52) kilograms, made up predominantly of hydrogen-1 (otherwise known as a proton plus an electron) and helium-4 (otherwise known as an alpha particle plus two electrons), 75% hydrogen-1 (by mass!) and 25% helium-4 (by mass!), corresponding to 15 sedecillion (1.5x10^52) kilograms of hydrogen (7.5 septedecillion moles of molecular hydrogen H_2 or 0.75x10^55 moles of H_2) and 5 sedecillion (0.5x10^52) kilograms of helium-4 (1.25 septedecillion moles of helium-4 or 0.125x10^55 moles of He-4; as the names suggest, helium-4 is almost four times as massive as hydrogen-1, so while there are only one-fourth as many helium-4's as there are hydrogen-1's, each helium-4 has four times the mass of each hydrogen-1, which would lead to exact mass equality for helium-4 and hydrogen-1 were it not for the binding energy of the two protons and the two neutrons that constitute the helium-4 nucleus, which is an alpha particle, the binding energy corresponding to the energy released when four protons fuse together to make the alpha particle as takes place in the thermonuclear reactions at the center of the Sun, for example, in strict accord with Einstein's justifiably famous "E=mc^2" equation, "E" being the energy released when a rest mass "m" is convereted to pure energy, the constant of proportionality being none other than "c" the speed of light in an absolute vacuum, 2.99792458x10^8 meters per second by definition, two-hundred ninety-nine million, seven-hundred ninety-two thousand, four hundred fifty-eight meters per second--the solar power density at one Astronomical Unit or AU, defined as the mean distance from the center of the Earth to the center of the Sun, 15 trillion (1.5x10^13) centimeters, is 1.4 kilowatts per square meter, meaning that the total luminosity of our Sun is 1.4 kilowatts per square meter times the surface area of a sphere cented at the center of the Sun having a radius of 1 AU, (4)x(pi)x(1.5x10^11 meters)^2=(4)x(22/7)x(9/4)x10^22 square meters=2.8x10^23 square meters or two-hundred eighty sextillion square meters, yielding a total luminosity L=(1.4x10^3 watts per square meter)x(2.8x10^23 square meters)=4x10^26 watts(!), which is a lot of watts, 400 septillion of them, which comes about when 4 billion (4x10^9) kilograms of hydrogen-1 rest mass is converted into pure energy by thermonuclear fusion processes at the center of our Sun every second, since E=mc^2=(4x10^9 kilograms)x(3x10^8 meters per second)^2=(4x10^9 kilograms)x(9x10^16 meters squared per second squared)=4x10^26 Joules (400 septillion Joules) every second is 4x10^26 watts(!!), 400 septillion of them, just what the good Dr. Einstein ordered!!!)--WOW, these have to be two of the longest parentheticals (at around 30 lines each!) ever to appear in a single (very long) paragraph in a blog! Could be the effect of the notorious "chemo brain" of which I've heard!
Well, now that we've gotten the swing of this number-naming thing, we can usefully apply our newfound skills to the numbers, prime and otherwise that occur in my investigations into Furlong's Conjecture. For example, Factorizer investigated the occurrence of twin primes, also known as prime pairs, between the squares of the twin primes 46,271 (46.271 thousand, also known as forty-six thousand, two-hundred seventy-one or 4.6271x10^4, 5 digit number) and 46,273 (46.273 thousand, also known as forty-six thousand, two-hundred seventy-three or 4.6273x10^4, a 5 digit number), or in other words (or numbers!), between 2,141,005,441=46,271^2 [2.141005411 billion, also known as two billion, one-hundred forty-one million, five thousand, four-hundred eleven or 2.141005411x10^9, a 10 digit number] through
2,141,190,529=46,273^2 [2.141190529 billion, also known as two billion, one-hundred forty-one million, one-hundred ninety thousand, four-hundred eleven 2.141190529x10^9, a 10 digit number]. I would bet any amount of money that this is the very first time in the long stretch of human history that either of the prime numbers 46,271 and 46,273 have ever had their proper English names written down! Similarly, my number theory kluge in Excel investigated the occurrence of twin primes, also known as prime pairs, between the squares of the twin primes 2,141,190,431 (2.141190431 billion, also known as two billion, one-hundred forty-one million, one-hundred ninety thousand, four-hundred thirty-one or 2.141190431x10^9, a 10 digit number) and 2,141,190,433 (2.141190433 billion, also known as two billion, one-hundred forty-one million, one-hundred ninety thousand, four-hundred thirty-three or 2.141190433x10^9, a 10 digit number), or in other words (or numbers!), between (2,141,190,431)^2=4,584,696,461,805,965,761 [4.584696461805965761 quintillion, also known as four quintillion, five-hundred eighty-four quadrillion, six-hundred ninety-six trillion, four-hundred sixty-one billion, eight-hundred five million, nine-hundred sixty-five thousand, seven-hundred sixty-one or 4.584696461805965761x10^18, a 19 digit number] and
(2,141,190,433)^2=4,584,696,470,370,727,489 [4.584696470370727489 quintillion, also known as four quintillion, five-hundred eighty-four quadrillion, six-hundred ninety-six trillion, four-hundred seventy billion, three-hundred seventy million, seven-hundred twenty-seven thousand, four-hundred eighty-nine or 4.584696470370727489x10^18, a 19 digit number]. Again, I would wager any amount of money that this is the very first time in the long stretch of human history that either of the prime numbers 2,141,190,431 and 2,141,190,433 have ever had their proper English names written down! You all are witnesses to a moment in time never before seen! Which is, of course, true of every moment in time, when you come to think about it, at least when "aided" by "chemo brain!"
Likewise, my Excel number theory kluge investigated the occurrence of twin primes, also known as prime pairs, between the squares of the twin primes 4,584,696,461,808,572,489 (4.584696461808572489 quintillion or 4.584696461808572489x10^18, a 19 digit number, also known as four quintillion, five-hundred eighty-four quadrillion, six-hundred ninety-six trillion, four-hundred sixty-one billion, eight-hundred eight million, five-hundred seventy-two thousand, four-hundred eighty-nine) and 4,584,696,461,808,572,491 (4.584696461808572491 quadrillion or 4.584696461808572491x10^18, a 19 digit number, also known as four quintillion, five-hundred eighty-four quadrillion, six-hundred ninety-six trillion, four-hundred sixty-one billion, eight-hundred eight million, five-hundred seventy-two thousand, four-hundred ninety-one), or in other words (or numbers!), found, between (4,584,696,461,808,572,489)^2=21,019,441,646,920,043,379,214,312,327,967,655,121
[21.019441646920043379214312327967655121 undecillion or 2.1019441646920043379214312327967655121x10^37, a 38 digit number, also known as twenty-one undecillion, nineteen decillion, four-hundred forty-one nonillion, six-hundred forty-six octillion, nine-hundred twenty septillion, forty-three sextillion, three-hundred seventy-nine quintillion, two-hundred fourteen quadrillion, three-hundred twelve trillion, three-hundred twenty-seven billion, nine-hundred sixty-seven million, six-hundred fifty-five thousand, one-hundred twenty-one] and (4,584,696,461,808,572,491)^2 =21,019,441,646,920,043,397,553,098,175,201,945,081 [21.019441646920043397553098175201945081
undecillion or 2.1019441646920043397553098175201945081x10^37, a 38 digit number, also known as twenty-one undecillion, nineteen decillion, four-hundred forty-one nonillion, six-hundred forty-six octillion, nine-hundred twenty septillion, forty-three sextillion, three-hundred ninety-seven quintillion, five-hundred fifty-three quadrillion, ninety-eight trillion, one-hundred seventy-five billion, two-hundred one million, nine-hundred forty-five thousand, eighty-one].
Last, but certainly not least by any stretch of the imagination(!), between (21,019,441,646,920,043,379,214,312,327,972,877,051)^2 =
441,816,927,148,276,785,559,214,254,431,324,696,206,279,105,710,628,344,677,315,905,110,362,456,601 [441.816927148276785559214254431324696206279105710628344677315905110362456601 tresvigintillion {because there are 24 clusters of three numbers in this leviathan number, the appropriate prefix according to Conway and Guy is related to 23} or 4.41816927148276785559214254431324696206279105710628344677315905110362456601x10^74, a 75 digit number, also known as four-hundred forty-one tresvigintillion, eight-hundred sixteen duovigintillion, nine-hundred twenty-seven unvigintillion, one-hundred forty-eight vigintillion, two-hundred seventy-six novendecillion, seven-hundred eight-five octodecillion, five-hundred fifty-nine septendecillion, two-hundred fourteen sedecillion, two-hundred fifty-four quinquadecillion, four-hundred thirty-one quattuordecillion, three-hundred twenty-four tredecillion, six-hundred ninety-six duodecillion, two-hundred six undecillion, two-hundred seventy-nine decillion, one-hundred five nonillion, seven-hundred ten octillion, six-hundred twenty-eight septillion, three-hundred forty-four sextillion, six-hundred seventy-seven quintillion, three-hundred fifteen quadrillion, nine-hundred five trillion, one-hundred ten billion, three-hundred sixty-two million, four-hundred fifty-six thousand, six-hundred one] and (21,019,441,646,920,043,379,214,312,327,972,877,053)^2 =
441,816,927,148,276,785,559,214,254,431,325,087,504,204,804,774,155,892,130,954,202,653,528,276,881 [441.816927148276785559214254431325087504204804774155892130954202653528276881 tresvigintillion or 441.816927148276785559214254431325087504204804774155892130954202653528276881x10^74, a 75 digit number, also known as four-hundred forty-one tresvigintillion, eight-hundred sixteen duovigintillion, nine-hundred twenty-seven unvigintillion, one-hundred forty-eight vigintillion, two-hundred seventy-six novendecillion, seven-hundred eight-five octodecillion, five-hundred fifty-nine septendecillion, two-hundred fourteen sedecillion, two-hundred fifty-four quinquadecillion, four-hundred thirty-one quattuordecillion, three-hundred twenty-five tredecillion, eighty-seven duodecillion, five-hundred four undecillion, two-hundred four decillion, eight-hundred four nonillion, seven-hundred seventy-four octillion, one-hundred fifty-five septillion, eight-hundred ninety-two sextillion, one-hundred thirty quintillion, five-hundred fifty-four quadrillion, two-hundred two trillion, six-hundred fifty-three billion, five-hundred twenty-eight million, two-hundred seventy-six thousand, eight-hundred eighty-one] there are at least 459 twin primes between (21,019,441,646,920,043,379,214,312,327,972,877,051)^2 =
441,816,927,148,276,785,559,214,254,431,324,696,206,279,105,710,628,344,677,315,905,110,362,456,601 [441.816927148276785559214254431324696206279105710628344677315905110362456601 tresvigintillion or 4.41816927148276785559214254431324696206279105710628344677315905110362456601x10^74, a 75 digit number] and
441,816,927,148,276,785,559,214,254,431,324,696,206,279,105,710,628,344,677,315,905,110,372,940,363 [441.816927148276785559214254431324696206279105710628344677315905110362456601 tresvigintillion] (with 391,297,925,699,063,527,547,453,638,297,543,165,820,280 - 10,483,762 =
391,297,925,699,063,527,547,453,638,297,543,155,336,518 [391.297925699063527547453638297543155336518 duodecillion or 3.91297925699063527547453638297543155336518x10^41, a 42 digit number, also known as three-hundred ninety-one duodecillion, two-hundred ninety-seven undecillion, nine-hundred twenty-five decillion, six-hundred ninety-nine nonillion, sixty-three octillion, five-hundred twenty-seven septillion, five-hundred forty-seven sextillion, four-hundred fifty-three quintillion, six-hundred thirty-eight quadrillion, two-hundred ninety-seven trillion, five-hundred forty-three billion, one-hundred fifty-five million, three-hundred thirty-six thousand, five-hundred eighteen] in the range to spare)!!! Thus, my number theory kluge in Excel further investigated the occurrence of twin primes, also known as prime pairs, between the squares of the twin primes 21,019,441,646,920,043,379,214,312,327,972,877,051 (21.019441646920043379214312327972877051 undecillion or 2.1019441646920043379214312327972877051x10^37, a 38 digit number, also known as twenty-one undecillion, nineteen decillion, four-hundred forty-one nonillion, six-hundred forty-six octillion, nine-hundred twenty septillion, forty-three sextillion, three-hundred seventy-nine quintillion, two-hundred fourteen quadrillion, three-hundred twelve trillion, three-hundred twenty-seven billion, nine-hundred seventy-two million, eight-hundred seventy-seven thousand, fifty-one, named here in English for the first time ever!) and 21,019,441,646,920,043,379,214,312,327,972,877,053 (21.019441646920043379214312327972877053 undecillion 2.1019441646920043379214312327972877053x10^37, a 38 digit number, also known as twenty-one undecillion, nineteen decillion, four-hundred forty-one nonillion, six-hundred forty-six octillion, nine-hundred twenty septillion, forty-three sextillion, three-hundred seventy-nine quintillion, two-hundred fourteen quadrillion, three-hundred twelve trillion, three-hundred twenty-seven billion, nine-hundred seventy-two million, eight-hundred seventy-seven thousand, fifty-three, also named here in English for the first time ever!), or in other words (or numbers!), between (21,019,441,646,920,043,379,214,312,327,972,877,051)^2 =
The next step is to meet again with my surgical oncologist, Dr. Paul Mansfield, to discuss whether it makes sense to pursue surgical options, such as Hyperthermal Intraperitoneal Chemotherapy (HIPEC), at this time. Hopefully, Dr. Mansfield will want to do the surgery! It's not any easy surgery to endure. It can take up to 14 hours! That's a long time to be under general anesthesia! The HIPEC part of the surgery itself includes 90 minutes or so of my being gently rocked back and forth while the warm chemo bath is sloshing around inside my abdominal cavity! The direct application of the chemo to the affected tissues is much more effective than chemo delivered intravenously, which, because it's necessarily delivered systemically throughout the whole body cannot be nearly as concentrated or as strong, because of the deliterious consequences of the deadly chemo cocktail. Supposedly, one-third of the patients who under go the full HIPEC treatment experience a complete remission of their cancers with no reoccurence of the cancers! I intend to be in that one-third! Those are good odds, indeed! Much better than the odds of winning the lottery, for example!
Needless to say, we're all very pleased by all this! Ecstatic, you might say! Euphoric! I'm still at MDACC getting my fifth FOLFOX6 chemo treatment, fifth out of twelve altogether. Thanks to Dave and Jo and Brian and Celeste, I'm able to post this in real time using our beloved Paddy the iPad! We love Paddy and can't imagine what it would be like living without it!
On an entirely unrelated note, for all of you who are wondering about the fate and robustness of Furlong's Conjecture (see previous posting of the same name--basically, I modestly conjectured that there will always be at least two twin prime pairs between the squares of successive prime numbers!), here's the latest and greatest that I've been able to coax out of Excel (equipped with an add-on that permits virtually unlimited precision integer operations and many very handy number theory operations to be performed--for more, see Joe Crump's Immortal Theory Number Theory website), although not without a lot of tsuris (Yiddish for trouble) and much kutput (Gujarati for trouble):
My nifty Factorizer program (not programmed by me, of course, but able to be purchased for a modest sum at Factorizer) found 531 prime pairs in the range from 2,141,005,441=46,271^2 [2.141005411 billion or 2.141005411x10^9, a 10 digit number] through 2,141,190,529=46,273^2 [2.141190529 billion or 2.141190529x10^9, a 10 digit number]. Needless to say, my conjecture is certainly robust enough (with 529=23^2 prime pairs to spare!) out to 46,271^2 and 46,273^2!!! Unfortunately, Factorizer does not do numbers larger than the Mersenne prime M_31=2^31-1=2,147,483,647 so I was forced to turn to my makeshift Excel number theory kluge!
Between (2,141,190,431)^2=4,584,696,461,805,965,761 [4.584696461805965761 quintillion or 4.584696461805965761x10^18, a 19 digit number] and
(2,141,190,433)^2=4,584,696,470,370,727,489 [4.584696470370727489 quintillion or 4.584696470370727489x10^18, a 19 digit number] there are at least 1863 twin primes between (2,141,190,431)^2=4,584,696,461,805,965,761 and 4,584,696,461,808,572,881 (with 8,564,761,728 - 2,607,120 = 8,562,154,608 [8.562154608 billion or 8.562154608x10^9, a 10 digit number] numbers in the range to spare)!!! The range of numbers chosen is not completely arbirtary! It merely reflects the fact that Excel only permits 65,536=2^16 rows in any worksheet and I'm too lazy to start another column of prime numbers, so for each range of numbers I'm really only sampling the first 60,000 or so numbers in each range (for reasons too unimportant to go into it just so happens that I start each column of my primes between squares of twin primes at row 5,536 or thereabouts, so I don't even get the full 65,536 numbers that I could otherwise get in each of columns)!
Between (4,584,696,461,808,572,489)^2=21,019,441,646,920,043,379,214,312,327,967,655,121
[21.019441646920043379214312327967655121 undecillion or 2.1019441646920043379214312327967655121x10^37, a 38 digit number] and (4,584,696,461,808,572,491)^2 =21,019,441,646,920,043,397,553,098,175,201,945,081 [21.019441646920043397553098175201945081 undecillion or 2.1019441646920043397553098175201945081x10^37, a 38 digit number] there are at least 940 twin primes between (4,584,696,461,808,572,489)^2 =21,019,441,646,920,043,379,214,312,327,967,655,121 and
21,019,441,646,920,043,379,214,312,327,972,886,863 (with 18,338,785,847,234,289,960 -
5,231,742 = 18,338,785,847,229,058,218 [18.338785847229058218 quintillion or 1.8338785847229058218x10^19, a 20 digit number] in the range to spare)!!!
At this point, a little explanation of terminology may be in order! The issue is how to express large numbers in English! John Conway and Richard Guy have a very good explanation of how we should go about doing this for numbers as large as you like in their superb book The Book of Numbers. Among other things, Conway and Guy point out that what it really all boils down to is coming up with reasonable names for the Nth zillion numbers that are powers of 10 that go like 10^(3N+3), according to the American naming system and not the British naming system, which uses 10^6N instead! The first few of these are well known and beloved by all: one million (1,000,000=10^6, a 7 digit number) for N=1, one billion (1,000,000,000=10^9, a 10 digit number) for N=2, one trillion (1,000,000,000,000=10^12, a 13 digit number) for N=3, one quadrillion (1,000,000,000,000,000=10^15, a 16 digit number) for N=4, one quintillion (1,000,000,000,000,000,000=10^18, a 19 digit number) for N=5, one sextillion (1,000,000,000,000,000,000,000=10^21, a 22 digit number) for N=6, one septillion (1,000,000,000,000,000,000,000,000=10^24, a 25 digit number) for N=7, one octillion (1,000,000,000,000,000,000,000,000,000=10^27, a 28 digit number) for N=8, one nonillion (1,000,000,000,000,000,000,000,000,000,000=10^30, a 31 digit number) for N=9, and one decillion (1,000,000,000,000,000,000,000,000,000,000,000=10^33, a 34 digit number) for N=10. Notice that the number of triple zero clusters is always one more than the value of N! That's because of the "+3" in the (3N+3) exponent in 10^(3N+3)!
When would you ever have a chance to use such names in everyday conversation? Well, the federal budget is in the trillions (a few times 10^12), for example. The mass of the Earth's atmosphere is roughly 4 quintillion (4x10^18) kilograms. The moon's mass is about 70 sextillion (7x10^22) kilograms. Avogadro's number is 602.214179(30) sextillion (6.02214179(30)x10^23), which is the number of carbon-12 atoms in 12 grams of pure carbon-12, for example (the "(30)" at the end of the number 6.02214179 reflects a "standard experimental uncertainty" of +/-30 in the last two digits [it understood that the number "(30)" in parentheses is the numerical value of the standard uncertainty, the estimated standard deviation, where the N data points are assumed to be adequately characterized by a Gaussian or normal probability distribution that looks like a Bell curve when plotted out, centered on the average or mean value X_average=Sum from i=1 to i=N of (X_i)/N, with a spread about the average value given by s=the square root of the Sum from i=1 to i=N of {(X_i-X_average)^2}/(N-1), the variance normalized by one less than the number N of data points {for exceedingly obscure reasons related to something called the "Student's t-distribution" that I won't go into, which is a welcome relief to all concerned!}] referred to the corresponding last digits of the quoted result--this implies that it is believed with an approximate level of confidence of 68.27% that the "true" value of Avogadro's number probably lies between 6.012214149x10^23 and 6.012214209x10^23, that it is believed with an approximate level of confidence of 95% that the "true" value of Avogadro's number probably lies between 6.012214119x10^23 and 6.012214239x10^23 [from an "expanded" experimental uncertainty of about +/-60 in the last two digits that may readily obtained by multiplying the quoted value of the "standard experimental uncertainty" by 1.960, which I've conveniently rounded up to 2.000, keeping 3 significant digits in the rounding and technically boosting my level of confidence from 95% to 95.45%, as may be confirmed here, at page 82, Table G.1], and that it is believed with an approximate level of confidence of 99% that the "true" value of Avogadro's number probably lies between 6.012214089x10^23 and 6.012214269x10^23 [from an "expanded" experimental uncertainty of about +/-90 in the last two digits that may readily obtained by multiplying the quoted value of the "standard experimental uncertainty" by 2.576, which I've conveniently rounded up to 3.000, keeping 3 significant digits in the rounding and technically boosting my level of confidence from 99% to 99.73%]). As pointed out in the marveously useful book Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin, if you had an Avogadro's number of fairly fat cats (at 10 kilograms each!), they would have a mass of just about 6 septillion (6x10^24) kilograms, which happens to be the mass of the entire Earth! The mass of Jupiter is in the ballpark of one octillion (10^27) kilograms, while the mass of our Sun is approximately 2 nonillion (2x10^30) kilograms or 2 decillion (2x10^33) grams! There are in the neighborhood of 100 billion (10^11) stars in the Milky Way galaxy so it has a mass of 200 duodecillion (2x10^41) kilograms or 200 tredecillion (2x10^44) grams. There are on the order of 100 billion (10^11) galaxies in the whole known Universe, as far as we know, so that the whole known Universe should have a mass of about 20 sedecillion (2x10^52) kilograms, made up predominantly of hydrogen-1 (otherwise known as a proton plus an electron) and helium-4 (otherwise known as an alpha particle plus two electrons), 75% hydrogen-1 (by mass!) and 25% helium-4 (by mass!), corresponding to 15 sedecillion (1.5x10^52) kilograms of hydrogen (7.5 septedecillion moles of molecular hydrogen H_2 or 0.75x10^55 moles of H_2) and 5 sedecillion (0.5x10^52) kilograms of helium-4 (1.25 septedecillion moles of helium-4 or 0.125x10^55 moles of He-4; as the names suggest, helium-4 is almost four times as massive as hydrogen-1, so while there are only one-fourth as many helium-4's as there are hydrogen-1's, each helium-4 has four times the mass of each hydrogen-1, which would lead to exact mass equality for helium-4 and hydrogen-1 were it not for the binding energy of the two protons and the two neutrons that constitute the helium-4 nucleus, which is an alpha particle, the binding energy corresponding to the energy released when four protons fuse together to make the alpha particle as takes place in the thermonuclear reactions at the center of the Sun, for example, in strict accord with Einstein's justifiably famous "E=mc^2" equation, "E" being the energy released when a rest mass "m" is convereted to pure energy, the constant of proportionality being none other than "c" the speed of light in an absolute vacuum, 2.99792458x10^8 meters per second by definition, two-hundred ninety-nine million, seven-hundred ninety-two thousand, four hundred fifty-eight meters per second--the solar power density at one Astronomical Unit or AU, defined as the mean distance from the center of the Earth to the center of the Sun, 15 trillion (1.5x10^13) centimeters, is 1.4 kilowatts per square meter, meaning that the total luminosity of our Sun is 1.4 kilowatts per square meter times the surface area of a sphere cented at the center of the Sun having a radius of 1 AU, (4)x(pi)x(1.5x10^11 meters)^2=(4)x(22/7)x(9/4)x10^22 square meters=2.8x10^23 square meters or two-hundred eighty sextillion square meters, yielding a total luminosity L=(1.4x10^3 watts per square meter)x(2.8x10^23 square meters)=4x10^26 watts(!), which is a lot of watts, 400 septillion of them, which comes about when 4 billion (4x10^9) kilograms of hydrogen-1 rest mass is converted into pure energy by thermonuclear fusion processes at the center of our Sun every second, since E=mc^2=(4x10^9 kilograms)x(3x10^8 meters per second)^2=(4x10^9 kilograms)x(9x10^16 meters squared per second squared)=4x10^26 Joules (400 septillion Joules) every second is 4x10^26 watts(!!), 400 septillion of them, just what the good Dr. Einstein ordered!!!)--WOW, these have to be two of the longest parentheticals (at around 30 lines each!) ever to appear in a single (very long) paragraph in a blog! Could be the effect of the notorious "chemo brain" of which I've heard!
Well, now that we've gotten the swing of this number-naming thing, we can usefully apply our newfound skills to the numbers, prime and otherwise that occur in my investigations into Furlong's Conjecture. For example, Factorizer investigated the occurrence of twin primes, also known as prime pairs, between the squares of the twin primes 46,271 (46.271 thousand, also known as forty-six thousand, two-hundred seventy-one or 4.6271x10^4, 5 digit number) and 46,273 (46.273 thousand, also known as forty-six thousand, two-hundred seventy-three or 4.6273x10^4, a 5 digit number), or in other words (or numbers!), between 2,141,005,441=46,271^2 [2.141005411 billion, also known as two billion, one-hundred forty-one million, five thousand, four-hundred eleven or 2.141005411x10^9, a 10 digit number] through
2,141,190,529=46,273^2 [2.141190529 billion, also known as two billion, one-hundred forty-one million, one-hundred ninety thousand, four-hundred eleven 2.141190529x10^9, a 10 digit number]. I would bet any amount of money that this is the very first time in the long stretch of human history that either of the prime numbers 46,271 and 46,273 have ever had their proper English names written down! Similarly, my number theory kluge in Excel investigated the occurrence of twin primes, also known as prime pairs, between the squares of the twin primes 2,141,190,431 (2.141190431 billion, also known as two billion, one-hundred forty-one million, one-hundred ninety thousand, four-hundred thirty-one or 2.141190431x10^9, a 10 digit number) and 2,141,190,433 (2.141190433 billion, also known as two billion, one-hundred forty-one million, one-hundred ninety thousand, four-hundred thirty-three or 2.141190433x10^9, a 10 digit number), or in other words (or numbers!), between (2,141,190,431)^2=4,584,696,461,805,965,761 [4.584696461805965761 quintillion, also known as four quintillion, five-hundred eighty-four quadrillion, six-hundred ninety-six trillion, four-hundred sixty-one billion, eight-hundred five million, nine-hundred sixty-five thousand, seven-hundred sixty-one or 4.584696461805965761x10^18, a 19 digit number] and
(2,141,190,433)^2=4,584,696,470,370,727,489 [4.584696470370727489 quintillion, also known as four quintillion, five-hundred eighty-four quadrillion, six-hundred ninety-six trillion, four-hundred seventy billion, three-hundred seventy million, seven-hundred twenty-seven thousand, four-hundred eighty-nine or 4.584696470370727489x10^18, a 19 digit number]. Again, I would wager any amount of money that this is the very first time in the long stretch of human history that either of the prime numbers 2,141,190,431 and 2,141,190,433 have ever had their proper English names written down! You all are witnesses to a moment in time never before seen! Which is, of course, true of every moment in time, when you come to think about it, at least when "aided" by "chemo brain!"
Likewise, my Excel number theory kluge investigated the occurrence of twin primes, also known as prime pairs, between the squares of the twin primes 4,584,696,461,808,572,489 (4.584696461808572489 quintillion or 4.584696461808572489x10^18, a 19 digit number, also known as four quintillion, five-hundred eighty-four quadrillion, six-hundred ninety-six trillion, four-hundred sixty-one billion, eight-hundred eight million, five-hundred seventy-two thousand, four-hundred eighty-nine) and 4,584,696,461,808,572,491 (4.584696461808572491 quadrillion or 4.584696461808572491x10^18, a 19 digit number, also known as four quintillion, five-hundred eighty-four quadrillion, six-hundred ninety-six trillion, four-hundred sixty-one billion, eight-hundred eight million, five-hundred seventy-two thousand, four-hundred ninety-one), or in other words (or numbers!), found, between (4,584,696,461,808,572,489)^2=21,019,441,646,920,043,379,214,312,327,967,655,121
[21.019441646920043379214312327967655121 undecillion or 2.1019441646920043379214312327967655121x10^37, a 38 digit number, also known as twenty-one undecillion, nineteen decillion, four-hundred forty-one nonillion, six-hundred forty-six octillion, nine-hundred twenty septillion, forty-three sextillion, three-hundred seventy-nine quintillion, two-hundred fourteen quadrillion, three-hundred twelve trillion, three-hundred twenty-seven billion, nine-hundred sixty-seven million, six-hundred fifty-five thousand, one-hundred twenty-one] and (4,584,696,461,808,572,491)^2 =21,019,441,646,920,043,397,553,098,175,201,945,081 [21.019441646920043397553098175201945081
undecillion or 2.1019441646920043397553098175201945081x10^37, a 38 digit number, also known as twenty-one undecillion, nineteen decillion, four-hundred forty-one nonillion, six-hundred forty-six octillion, nine-hundred twenty septillion, forty-three sextillion, three-hundred ninety-seven quintillion, five-hundred fifty-three quadrillion, ninety-eight trillion, one-hundred seventy-five billion, two-hundred one million, nine-hundred forty-five thousand, eighty-one].
Last, but certainly not least by any stretch of the imagination(!), between (21,019,441,646,920,043,379,214,312,327,972,877,051)^2 =
441,816,927,148,276,785,559,214,254,431,324,696,206,279,105,710,628,344,677,315,905,110,362,456,601 [441.816927148276785559214254431324696206279105710628344677315905110362456601 tresvigintillion {because there are 24 clusters of three numbers in this leviathan number, the appropriate prefix according to Conway and Guy is related to 23} or 4.41816927148276785559214254431324696206279105710628344677315905110362456601x10^74, a 75 digit number, also known as four-hundred forty-one tresvigintillion, eight-hundred sixteen duovigintillion, nine-hundred twenty-seven unvigintillion, one-hundred forty-eight vigintillion, two-hundred seventy-six novendecillion, seven-hundred eight-five octodecillion, five-hundred fifty-nine septendecillion, two-hundred fourteen sedecillion, two-hundred fifty-four quinquadecillion, four-hundred thirty-one quattuordecillion, three-hundred twenty-four tredecillion, six-hundred ninety-six duodecillion, two-hundred six undecillion, two-hundred seventy-nine decillion, one-hundred five nonillion, seven-hundred ten octillion, six-hundred twenty-eight septillion, three-hundred forty-four sextillion, six-hundred seventy-seven quintillion, three-hundred fifteen quadrillion, nine-hundred five trillion, one-hundred ten billion, three-hundred sixty-two million, four-hundred fifty-six thousand, six-hundred one] and (21,019,441,646,920,043,379,214,312,327,972,877,053)^2 =
441,816,927,148,276,785,559,214,254,431,325,087,504,204,804,774,155,892,130,954,202,653,528,276,881 [441.816927148276785559214254431325087504204804774155892130954202653528276881 tresvigintillion or 441.816927148276785559214254431325087504204804774155892130954202653528276881x10^74, a 75 digit number, also known as four-hundred forty-one tresvigintillion, eight-hundred sixteen duovigintillion, nine-hundred twenty-seven unvigintillion, one-hundred forty-eight vigintillion, two-hundred seventy-six novendecillion, seven-hundred eight-five octodecillion, five-hundred fifty-nine septendecillion, two-hundred fourteen sedecillion, two-hundred fifty-four quinquadecillion, four-hundred thirty-one quattuordecillion, three-hundred twenty-five tredecillion, eighty-seven duodecillion, five-hundred four undecillion, two-hundred four decillion, eight-hundred four nonillion, seven-hundred seventy-four octillion, one-hundred fifty-five septillion, eight-hundred ninety-two sextillion, one-hundred thirty quintillion, five-hundred fifty-four quadrillion, two-hundred two trillion, six-hundred fifty-three billion, five-hundred twenty-eight million, two-hundred seventy-six thousand, eight-hundred eighty-one] there are at least 459 twin primes between (21,019,441,646,920,043,379,214,312,327,972,877,051)^2 =
441,816,927,148,276,785,559,214,254,431,324,696,206,279,105,710,628,344,677,315,905,110,362,456,601 [441.816927148276785559214254431324696206279105710628344677315905110362456601 tresvigintillion or 4.41816927148276785559214254431324696206279105710628344677315905110362456601x10^74, a 75 digit number] and
441,816,927,148,276,785,559,214,254,431,324,696,206,279,105,710,628,344,677,315,905,110,372,940,363 [441.816927148276785559214254431324696206279105710628344677315905110362456601 tresvigintillion] (with 391,297,925,699,063,527,547,453,638,297,543,165,820,280 - 10,483,762 =
391,297,925,699,063,527,547,453,638,297,543,155,336,518 [391.297925699063527547453638297543155336518 duodecillion or 3.91297925699063527547453638297543155336518x10^41, a 42 digit number, also known as three-hundred ninety-one duodecillion, two-hundred ninety-seven undecillion, nine-hundred twenty-five decillion, six-hundred ninety-nine nonillion, sixty-three octillion, five-hundred twenty-seven septillion, five-hundred forty-seven sextillion, four-hundred fifty-three quintillion, six-hundred thirty-eight quadrillion, two-hundred ninety-seven trillion, five-hundred forty-three billion, one-hundred fifty-five million, three-hundred thirty-six thousand, five-hundred eighteen] in the range to spare)!!! Thus, my number theory kluge in Excel further investigated the occurrence of twin primes, also known as prime pairs, between the squares of the twin primes 21,019,441,646,920,043,379,214,312,327,972,877,051 (21.019441646920043379214312327972877051 undecillion or 2.1019441646920043379214312327972877051x10^37, a 38 digit number, also known as twenty-one undecillion, nineteen decillion, four-hundred forty-one nonillion, six-hundred forty-six octillion, nine-hundred twenty septillion, forty-three sextillion, three-hundred seventy-nine quintillion, two-hundred fourteen quadrillion, three-hundred twelve trillion, three-hundred twenty-seven billion, nine-hundred seventy-two million, eight-hundred seventy-seven thousand, fifty-one, named here in English for the first time ever!) and 21,019,441,646,920,043,379,214,312,327,972,877,053 (21.019441646920043379214312327972877053 undecillion 2.1019441646920043379214312327972877053x10^37, a 38 digit number, also known as twenty-one undecillion, nineteen decillion, four-hundred forty-one nonillion, six-hundred forty-six octillion, nine-hundred twenty septillion, forty-three sextillion, three-hundred seventy-nine quintillion, two-hundred fourteen quadrillion, three-hundred twelve trillion, three-hundred twenty-seven billion, nine-hundred seventy-two million, eight-hundred seventy-seven thousand, fifty-three, also named here in English for the first time ever!), or in other words (or numbers!), between (21,019,441,646,920,043,379,214,312,327,972,877,051)^2 =
441,816,927,148,276,785,559,214,254,431,324,696,206,279,105,710,628,344,677,315,905,110,362,456,601 [441.816927148276785559214254431324696206279105710628344677315905110362456601 tresvigintillion or 4.41816927148276785559214254431324696206279105710628344677315905110362456601x10^74, a 75 digit number, also known as four-hundred forty-one tresvigintillion, eight-hundred sixteen duovigintillion, nine-hundred twenty-seven unvigintillion, one-hundred forty-eight vigintillion, two-hundred seventy-six novendecillion, seven-hundred eight-five octodecillion, five-hundred fifty-nine septendecillion, two-hundred fourteen sedecillion, two-hundred fifty-four quinquadecillion, four-hundred thirty-one quattuordecillion, three-hundred twenty-four tredecillion, six-hundred ninety-six duodecillion, two-hundred six undecillion, two-hundred seventy-nine decillion, one-hundred five nonillion, seven-hundred ten octillion, six-hundred twenty-eight septillion, three-hundred forty-four sextillion, six-hundred seventy-seven quintillion, three-hundred fifteen quadrillion, nine-hundred five trillion, one-hundred ten billion, three-hundred sixty-two million, four-hundred fifty-six thousand, six-hundred one] and (21,019,441,646,920,043,379,214,312,327,972,877,053)^2 =
441,816,927,148,276,785,559,214,254,431,325,087,504,204,804,774,155,892,130,954,202,653,528,276,881 [441.816927148276785559214254431325087504204804774155892130954202653528276881 tresvigintillion or 441.816927148276785559214254431325087504204804774155892130954202653528276881x10^74, a 75 digit number, also known as four-hundred forty-one tresvigintillion, eight-hundred sixteen duovigintillion, nine-hundred twenty-seven unvigintillion, one-hundred forty-eight vigintillion, two-hundred seventy-six novendecillion, seven-hundred eight-five octodecillion, five-hundred fifty-nine septendecillion, two-hundred fourteen sedecillion, two-hundred fifty-four quinquadecillion, four-hundred thirty-one quattuordecillion, three-hundred twenty-five tredecillion, eighty-seven duodecillion, five-hundred four undecillion, two-hundred four decillion, eight-hundred four nonillion, seven-hundred seventy-four octillion, one-hundred fifty-five septillion, eight-hundred ninety-two sextillion, one-hundred thirty quintillion, five-hundred fifty-four quadrillion, two-hundred two trillion, six-hundred fifty-three billion, five-hundred twenty-eight million, two-hundred seventy-six thousand, eight-hundred eighty-one]. Finally, for the sake of completeness, the two largest twin primes found to date in my investigations of Furlong's Conjecture:
441,816,927,148,276,785,559,214,254,431,324,696,206,279,105,710,628,344,677,315,905,110,372,931,057 [441.816927148276785559214254431324696206279105710628344677315905110372931057 tresvigintillion or 4.41816927148276785559214254431324696206279105710628344677315905110372931057x10^74, a 75 digit number, also known as four-hundred forty-one tresvigintillion, eight-hundred sixteen duovigintillion, nine-hundred twenty-seven unvigintillion, one-hundred forty-eight vigintillion, two-hundred seventy-six novendecillion, seven-hundred eight-five octodecillion, five-hundred fifty-nine septendecillion, two-hundred fourteen sedecillion, two-hundred fifty-four quinquadecillion, four-hundred thirty-one quattuordecillion, three-hundred twenty-four tredecillion, six-hundred ninety-six duodecillion, two-hundred six undecillion, two-hundred seventy-nine decillion, one-hundred five nonillion, seven-hundred ten octillion, six-hundred twenty-eight septillion, three-hundred forty-four sextillion, six-hundred seventy-seven quintillion, three-hundred fifteen quadrillion, nine-hundred five trillion, one-hundred ten billion, three-hundred seventy-two million, nine-hundred thirty-one thousand, fifty-seven, named here in English for the first time ever!] and
441,816,927,148,276,785,559,214,254,431,324,696,206,279,105,710,628,344,677,315,905,110,372,931,059 [441.816927148276785559214254431324696206279105710628344677315905110372931059 tresvigintillion or 441.816927148276785559214254431324696206279105710628344677315905110372931059x10^74, a 75 digit number, also known as four-hundred forty-one tresvigintillion, eight-hundred sixteen duovigintillion, nine-hundred twenty-seven unvigintillion, one-hundred forty-eight vigintillion, two-hundred seventy-six novendecillion, seven-hundred eight-five octodecillion, five-hundred fifty-nine septendecillion, two-hundred fourteen sedecillion, two-hundred fifty-four quinquadecillion, four-hundred thirty-one quattuordecillion, three-hundred twenty-four tredecillion, six-hundred ninety-six duodecillion, two-hundred six undecillion, two-hundred seventy-nine decillion, one-hundred five nonillion, seven-hundred ten octillion, six-hundred twenty-eight septillion, three-hundred forty-four sextillion, six-hundred seventy-seven quintillion, three-hundred fifteen quadrillion, nine-hundred five trillion, one-hundred ten billion, three-hundred seventy-two million, nine-hundred thirty-one thousand, fifty-nine, also named here in English for the first time ever!]
My Excel kluge number theory program is supposed to be able to handle numbers up to 250 digits in length, so all that remains to be done is to square these last two twin primes, generating a range of numbers with about 150 digits each. Based on the trends I've observed up to now, I expect to find around 250 or so twin primes in the first 60,000 or so of this range. Furlong's Conjecture appears to be in good health, too!
Thursday, March 3, 2011
An Important CT Scan Today!
This evening sometime around 6:00PM, hopefully, I will have a CT scan that will show whether or not the 4 rounds of chemotherapy I've had have had an effect on the tumors in my abdominal cavity, both the primary non-carcinoid, mucinous appendiceal adenocarcinoma in and around what's left of my appendix and any and all metastatic nodules on and/or in my peritoneum! The CT scans (technically, there are 2 CT scans, one of my chest and the other of my abdomen/pelvis, but they'll do them both during the same session) will also help determine whether any rogue cells have escaped from my abdominal cavity to anywhere else in my body that such cells are likely to go to (apparently adenocarcinomas from the gut don't often metastasize to the brain or toes, for example)! We're fervently praying and hoping for a good report so we can plan and schedule HIPEC surgery very soon! The sooner these cancer cells are excised from my body and/or killed dead as little doornails, the better! Stay tuned and thanks so much again for all your many wonderful prayers and best wishes and love--I don't know what I would do without all of you! Thank God I have you all on my side! We'll whip this impertinent appendix cancer sooner rather than later! May God bless you all!
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