Just a few notes to clarify Furlong's Postulate! A "twin prime" is usually taken to mean 2 prime numbers that are separated by 1 and only 1 even composite number, which must necessarily be some multiple of 6, except for the very first set of twin primes, namely, 3 and 5, which are separated by 1 and only 1 even composite number (4), which is manifestly not some multiple of 6. Just as I did before in the case of Bertrand's Postulate, we can easily verify Furlong's Postulate by "daisy chaining" up the number line: the 3rd twin prime pair (11, 13) is strictly between 7 and 2*7 = 14, the 4th twin prime pair (17, 19) is strictly between 13 and 2*13 = 26, the 5th twin prime pair (29, 31) is strictly between 19 and 2*19 = 38, the 7th twin prime pair (59, 61) is strictly between 31 and 2*31 = 62, the 10th twin prime pair (107, 109) is strictly between 61 and 2*61 = 122, the 15th twin prime pair (197, 199) is strictly between 109 and 2*109 = 218, and the 21st twin prime pair (347, 349) is strictly between 199 and 2*199 = 398, and so forth!
By the way, I have no earthly idea how anyone would go about trying to prove my postulate for all primes! It's not even known for sure whether there are an infinite number of twin primes! Almost every number theory nut worth their salt believes that there must be an infinite number of twin primes! Of course, if someone could prove that my postulate is true, somehow, then, since there are an infinite number of prime numbers, that would necessarily mean that there would have to be an infinite number of twin primes!
Randy and Caroline
A lovely July in Seattle!
Wednesday, August 22, 2012
And the Story Continues ...
My previous blog post wasn't meant to end when it did, although it did make it like a cliff-hanger! I'm having some editing issues writing my blog post using 'Nother Paddy! Anyway, this will boost my overall number of posts, at least!
As I was saying, the results of my Thursday, August 16, 2012 CT scan were a bit inconclusive. There was still no evidence of disease outside my abdominal cavity, thank God! However, the tumors definitely were not shrinking, but instead had gotten somewhat bigger than they were in my June 2012 CT scan, although it was hard to quantify just how much bigger, since my mucinous signet ring cell tumors are apparently rather poorly differentiated (not very encouraging, since the more poorly differentiated the tumors are the more aggressive they tend to be!) and are spread out somewhat amoeba-like on my peritoneum, the membrane that encompasses my abdominal cavity! The very experienced radiologist was able to report to Dr. Shureiqi that none of the tumors appeared to have grown by 25%, which is good news! If the tumors would have grown by 25% or more, then we would have definitely concluded that the FOLFIRI with Avastin was not being effective, in which case we would most likely stop using Irenitecan in the chemo cocktail and go back to FOLFOX, using Oxalyplatin coupled this time with Avastin. Last year, FOLFOX definitely was able to stop the tumor growth in its tracks!
The potential downside of returning to Oxalyplatin is that the cumulative peripheral neuropathy would probably worsen. Oxalyplatin was the chemo used in my 12 hour Cytoreduction surgery with HIPEC on Tuesday, April 12, 2011, last year, and the toes on both of my feet, especially my right foot, are still numb and tingly! Not to mention the inconvenience of having to forswear for a while my favorite Starbucks iced quad espresso drink, because of the weird effect Oxalyplatin has on one's throat!
I'm scheduled to have yet another CT scan in October, so we decided to continue with the current chemo regimen until then and reassess then what we should do! Please pray for wisdom and insight so that we'll have a better idea of what to do!
And now for something completely different! My latest number theory conjecture! A mathematician named Bertrand famously postulated that there always exists at least one prime number between any number n and 2 less than twice n, for all numbers n greater than or equal to 4. That's the stronger form of Bertrand's Postulate. The weaker form of Bertrand's Postulate is more well known and it says that there always exists at least one prime number between any number n and twice n (2*n), for all n greater than 1! There is a very nice proof of the weaker form of Bertrand's Postulate given in Wikipedia! We can easily "daisy chain" our way up the number line as far as we want to go verifying either form of Bertrand's Postulate!
For example, the 2nd prime number 3 is strictly between 2 and 2*2 = 4, the 3rd prime number 5 is strictly between 3 and 2*3 = 6, the 4th prime number 7 is strictly between 5 and 2*5 = 10, the 6th prime number 13 is strictly between 7 and 2*7 = 14, the 9th prime number 23 is strictly between 13 and 2*13 = 26, the 14th prime number 43 is strictly between 23 and 2*23 = 46, the 23rd prime number 83 is strictly between 43 and 2*43 = 86, the 38th prime number 163 is strictly between 83 and 2*83 = 166, the 66th prime number 317 is strictly between 163 and 2*163 = 326, the 115th prime number 631 is strictly between 317 and 2*317 = 634, the 205th prime number 1259 is strictly between 631 and 2*631 = 1262, the 368th prime number 2503 is strictly between 1259 and 2*1259 = 2518, the 670th prime number 5003 is strictly between 2503 and 2*2503 = 5006, and the 1229th prime number (1229 itself is a prime number and equals the total number of prime numbers less than 10000!) 9973 is strictly between 5003 and 2*5003 = 10006, and so forth! Furlong's Postulate is quite simply that there will always be at least 1 "twin prime" strictly between any prime number (p) and twice that prime number (2*p), for every prime number greater than 5
As I was saying, the results of my Thursday, August 16, 2012 CT scan were a bit inconclusive. There was still no evidence of disease outside my abdominal cavity, thank God! However, the tumors definitely were not shrinking, but instead had gotten somewhat bigger than they were in my June 2012 CT scan, although it was hard to quantify just how much bigger, since my mucinous signet ring cell tumors are apparently rather poorly differentiated (not very encouraging, since the more poorly differentiated the tumors are the more aggressive they tend to be!) and are spread out somewhat amoeba-like on my peritoneum, the membrane that encompasses my abdominal cavity! The very experienced radiologist was able to report to Dr. Shureiqi that none of the tumors appeared to have grown by 25%, which is good news! If the tumors would have grown by 25% or more, then we would have definitely concluded that the FOLFIRI with Avastin was not being effective, in which case we would most likely stop using Irenitecan in the chemo cocktail and go back to FOLFOX, using Oxalyplatin coupled this time with Avastin. Last year, FOLFOX definitely was able to stop the tumor growth in its tracks!
The potential downside of returning to Oxalyplatin is that the cumulative peripheral neuropathy would probably worsen. Oxalyplatin was the chemo used in my 12 hour Cytoreduction surgery with HIPEC on Tuesday, April 12, 2011, last year, and the toes on both of my feet, especially my right foot, are still numb and tingly! Not to mention the inconvenience of having to forswear for a while my favorite Starbucks iced quad espresso drink, because of the weird effect Oxalyplatin has on one's throat!
I'm scheduled to have yet another CT scan in October, so we decided to continue with the current chemo regimen until then and reassess then what we should do! Please pray for wisdom and insight so that we'll have a better idea of what to do!
And now for something completely different! My latest number theory conjecture! A mathematician named Bertrand famously postulated that there always exists at least one prime number between any number n and 2 less than twice n, for all numbers n greater than or equal to 4. That's the stronger form of Bertrand's Postulate. The weaker form of Bertrand's Postulate is more well known and it says that there always exists at least one prime number between any number n and twice n (2*n), for all n greater than 1! There is a very nice proof of the weaker form of Bertrand's Postulate given in Wikipedia! We can easily "daisy chain" our way up the number line as far as we want to go verifying either form of Bertrand's Postulate!
For example, the 2nd prime number 3 is strictly between 2 and 2*2 = 4, the 3rd prime number 5 is strictly between 3 and 2*3 = 6, the 4th prime number 7 is strictly between 5 and 2*5 = 10, the 6th prime number 13 is strictly between 7 and 2*7 = 14, the 9th prime number 23 is strictly between 13 and 2*13 = 26, the 14th prime number 43 is strictly between 23 and 2*23 = 46, the 23rd prime number 83 is strictly between 43 and 2*43 = 86, the 38th prime number 163 is strictly between 83 and 2*83 = 166, the 66th prime number 317 is strictly between 163 and 2*163 = 326, the 115th prime number 631 is strictly between 317 and 2*317 = 634, the 205th prime number 1259 is strictly between 631 and 2*631 = 1262, the 368th prime number 2503 is strictly between 1259 and 2*1259 = 2518, the 670th prime number 5003 is strictly between 2503 and 2*2503 = 5006, and the 1229th prime number (1229 itself is a prime number and equals the total number of prime numbers less than 10000!) 9973 is strictly between 5003 and 2*5003 = 10006, and so forth! Furlong's Postulate is quite simply that there will always be at least 1 "twin prime" strictly between any prime number (p) and twice that prime number (2*p), for every prime number greater than 5
Tumor and Tumor and Tumor Creeps in this Petty Pace ...
Summertime, and the living is easy! Since my last post, I've continued with my fortnightly chemo treatments (FOLFIRI with Avastin), having had 9 treatments this year, and 22 total (including the 5 FOLFOX and 8 FOLFIRI treatments last year)! Other than relatively minor nausea, I'm pleased to report that there continue to be very few side effects from the chemo! Prayer works! Please keep praying! And thanks so much for all your wonderful prayers and positive thoughts and good vibes! They are all truly life-giving and continue to buoy my spirits!
My spirits have needed extra buoying lately! When I met with my wonderful colorectal oncologist, Dr. Imad Shureiqi, on Friday morning, August 10, 2012, he showed me that one of the tumor markers they've been tracking ever since my diagnosis of appendiceal adenocarcinoma in December 2010, Carbohydrate Antigen 19-9 (CA 19-9), had trended upwards and, for the first time since my diagnosis, was above the normal range. The plot looked like the classic "hockey stick" graph of exponential growth, not unlike the one Vice President Al Gore showed in his film An Inconvenient Truth that tracked atmospheric concentrations of carbon dioxide! Somewhat alarming, to say the least!
Dr. Shureiqi ordered another CT scan, which I had last Thursday, August 16, 2012. Caroline and I met with Dr. Shureiqi the next day, Friday, August 17, 2012. The CA 19-9 level had increased even further, the slope seeming to approach the vertical asymptotically! And the results of the CT scan were inconclusive, at best.
My spirits have needed extra buoying lately! When I met with my wonderful colorectal oncologist, Dr. Imad Shureiqi, on Friday morning, August 10, 2012, he showed me that one of the tumor markers they've been tracking ever since my diagnosis of appendiceal adenocarcinoma in December 2010, Carbohydrate Antigen 19-9 (CA 19-9), had trended upwards and, for the first time since my diagnosis, was above the normal range. The plot looked like the classic "hockey stick" graph of exponential growth, not unlike the one Vice President Al Gore showed in his film An Inconvenient Truth that tracked atmospheric concentrations of carbon dioxide! Somewhat alarming, to say the least!
Dr. Shureiqi ordered another CT scan, which I had last Thursday, August 16, 2012. Caroline and I met with Dr. Shureiqi the next day, Friday, August 17, 2012. The CA 19-9 level had increased even further, the slope seeming to approach the vertical asymptotically! And the results of the CT scan were inconclusive, at best.
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