Furlong's Conjecture

This past Saturday, I re-entered the "prime" of my life, I turned 53, by the grace of God! As everyone knows, a prime number is a whole number (integer, conventionally denoted as p, for "prime") greater than 1 that has 2 and only 2 distinct whole number divisors, whole numbers that divide into another whole number without leaving any non-zero remainder, namely 1 and itself, p. Indeed, p divided by 1 equals p, and p divided by p equals 1. The first prime number p_1 is 2, which is the unique even prime number (every even number greater than 2 is divisible by 2, in addition to being divisible by 1 and itself, so it cannot be prime)! The next prime number p_2 is 3, since 3 is not evenly divisible by 2. The first composite number, a number greater than 1 that is not prime, is 4, which is evenly divisible by 2 exactly twice. The third prime number p_3 is 5, the second composite number is 6 (which also happens to be the first perfect number, a number that is equal to the sum of all its proper or aliquot divisors, all its divisors that are strictly less than the number itself: the proper divisors of 6 are 1, 2, and 3, and 1+2+3=6), the fourth prime number p_4 is 7, the third composite number is 8, which is evenly divisible by 2 exactly thrice, the fourth composite number is 9, which is (oddly) divisible by 3 exactly twice, and the fifth composite number is 10, which is evenly divisible by 2 exactly once and by 5 exactly once. One can proceed similarly forever!

There happen to be 4 prime numbers less than 10, 25 prime numbers less than 100, 168 prime numbers less than 1000, 1229 prime numbers less than 10000, 9592 prime numbers less than 100000, and 78498 prime numbers less than 1000000, for example! The first 25 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. Note that there is one unique pair of neighboring prime numbers, 2 and 3. There are also a fair number of "twin primes" separated by one and only one composite number, such as 3 and 5, 5 and 7, and 11 and 13, for example. It is not known whether there are an unlimited number of twin primes. Euclid cleverly proved centuries ago that there are an infinite number of prime numbers. Assume to the contrary that there are only a finite number of prime numbers, p_1, p_2, p_3, ..., p_n, where p_n is the "largest known prime number." Multiply all of the known primes together and then add 1, forming N=p_n#+1, where p_n# is "p_n primorial", the product of all the prime numbers less than or equal to p_n. Since N is (obviously) greater tha 1, N either is prime or composite. If N is prime, then since N is evidently (much, much) larger than p_n, the so-called "largest known prime," we have demonstrated that there is always at least one more prime number than we thought there were! If N is composite and not prime then N must necessarily have one or more prime divisors that are less than N. However, clearly none of the "known" prime numbers, p_1, p_2, p_3, ..., p_n, are able to divide N without leaving a remainder of 1, so there must exist at least one more prime number p that is greater than the so-called "largest prime number p_n, implying once again that our original assumption that there are only a finite number of prime numbers is wrong and there must therefore be an infinite or unlimited number of prime numbers!

Brocard's conjecture is that there are at least 4 prime numbers between the squares of successive odd prime numbers. I'm pleased to make "Furlong's conjecture," which necessarily includes Brocard's conjecture, but is much more restrictive: there are at least 2 twin primes between the squares of successive odd prime numbers. I have absolutely no idea of how to go about proving my conjecture! I have been able to verify the validity of my conjecture, by judicious sampling, for prime numbers up into the billions (and beyond)! I haven't found a counter-example, yet, but I'll keep looking and be sure to let you all know the outcome!

Arms and the Man, Four Rounds of Chemo Down So Far!

I'm pleased to report that my left arm is healing quite well!  We went in to see the orthopedist at M.D. Anderson Cancer Center on Monday, February 7, 2011, and the x-ray showed new bone being formed to knit together my left humerus across its surgical neck!  In about 4 weeks, I'll be able to start any needed physical therapy and stop wearing a sling!

I'm also pleased to report that I've survived a third and a fourth round of chemotherapy, with hardly any side effects worth reporting!  Hopefully, the chemo treatments are affecting the tumors much more than they are affecting me!  The neuropathy caused by the oxaliplatin part of my FOLFOX regimen does seem to be getting worse with each treatment, although the effects wear off over the course of the two weeks of each chemo treatment.  The numbness and tingling in my fingers and toes and the sensitivity to cold is somewhat annoying, of course, but the worst part, for me at least, is the weird feeling in your throat whenever you try to drink iced drinks or cold liquids.  It's not exactly like nurse Ida explained, that drinking cold beverages feels just like swallowing broken shards of glass, but it is very unpleasant, nevertheless!  Especially to someone like me who is hopelessly addicted to iced quad espressos from Starbucks!  Fortunately, room temperature drinks and hot beverages are fine!  If I have to, I may switch to a hot Venti Americano for a few days!

We are scheduled to meet with my colorectal oncologist, Dr. Imad Shureiqi, this coming Friday, February 25, 2011.  I think the plan is to allow this most recent (fourth) round of chemo to run it's course and then get another CT scan to ascertain whether or not the metastatic non-carcinoid, mucinous appendiceal adenocarcinoma has responded to the chemotherapy.  Hopefully, all the tumors, from the primary tumor in the appendix (or what's left of my appendix) to the metastatic nodules on my peritoneum, will have stopped growing and will have actively shrunk in size, or even better, will have died off altogether!

If so, then we will meet again with my surgical oncologist, Dr. Paul Mansfield, and discuss potential surgical options.  Since my appendix cancer is metastatic, as shown by the needle biopsy performed on one of the peritoneal nodules near the surface of my skin, the aggressive surgery that includes hyperthermal intraperitoneal chemotherapy (HIPEC) may well be in my future!  After my appendix and the right half of my colon and any involved lymph nodes and any and all peritoneal nodules have been removed from my abdominal cavity, heated chemotherapy chemicals are introduced into my abdominal cavity, bathing all the tissues directly with the chemo agents heated to about 40 degrees Centigrade (which is 104 degrees Fahrenheit).  Apparently, during the warm chemo bath, my body will be gently shaken for about 90 minutes (I will be anesthetized, of course, so I probably won't feel the gentle shaking!), insuring that all potentially affected portions within my peritoneum will be adequately exposed directly to the warm chemo.  This direct exposure of the tissues to the warm chemo is apparently much more effective than chemo delivered through my veins.  Roughly one-third of the patients receiving the HIPEC surgery treatment have complete remission of their cancers with no recurrence of the disease, or so I'm told.  Those are pretty good odds!  Much better than winning the lottery, for example!  After I've recovered adequately from the surgery, which could take up to six weeks or two months, especially after the HIPEC treatment, then I will continue with another eight rounds of the FOLFOX chemo treatments, for a grand total of twelve.

Thanks again for all your thoughts and prayers!  They are definitely working and very definitely appreciated!