20081223, 22:29  #1  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}×3×641 Posts 
A new Strong Law of Small Numbers example
From the NMBRTHRY mailing list comes a recent example of the first Strong Law of Small Numbers (http://mathworld.wolfram.com/StrongL...llNumbers.html).
Last Sunday: Quote:
Quote:
Last fiddled with by cheesehead on 20081223 at 22:34 

20081223, 22:54  #2 
Aug 2006
13541_{8} Posts 
Well, so much for sums of thin sequences representing all integers!
Kind of makes me wish I'd pushed ahead to find the counterexample myself. 
20081224, 08:13  #3 
"Robert Gerbicz"
Oct 2005
Hungary
2·733 Posts 
All counter example for the original conjecture up to 2^33 are:
Code:
Counterexample=3970902 Counterexample=39022919 Counterexample=102132857 Counterexample=110468517 Counterexample=368495972 Counterexample=391099413 Counterexample=395147912 Counterexample=421129348 Counterexample=452808398 Counterexample=776218485 Counterexample=1005771844 Counterexample=1485470432 Counterexample=3038310485 Counterexample=3263773338 Counterexample=3485976107 Counterexample=3640901241 Counterexample=3758331463 Counterexample=3784200441 Counterexample=3944795435 Counterexample=4014507719 Counterexample=4277741986 Counterexample=4397438442 Counterexample=4542739955 Counterexample=4757066466 Counterexample=5167438708 Counterexample=7130095749 Counterexample=7213669167 Counterexample=7424527675 Counterexample=7696559526 Counterexample=8309766941 Counterexample=8462583631 
20081226, 18:38  #4  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}×3×641 Posts 
Quote:
ZhiWei Sun isn't easily discouraged. From the NMBRTHRY mailing list comes: Quote:
Quote:
The clock is running ... Last fiddled with by cheesehead on 20081226 at 18:43 

20081226, 19:22  #5 
Aug 2006
3^{2}×5×7×19 Posts 

20081226, 20:23  #6 
"Robert Gerbicz"
Oct 2005
Hungary
2·733 Posts 

20090115, 20:08  #7 
Aug 2006
3^{2}·5·7·19 Posts 
I love when he (rarely) adds timestamps in addition to datastamps to his conjectures. Who else...
He's now offering ErdÅ‘sstyle prizes for proofs or counterexamples: http://listserv.nodak.edu/cgibin/wa...0&F=&S=&P=1395 Conjecture 1 seems considerably harder than the strong Goldbach conjecture, so I think his money is safe. The conjecture is almost certainly true. Conjecture 2 deals with a thin sequence (O(n log n) sums up to n), but it's thicker than Crocker's 1971 p + 2^a + 2^b sequence, so it may hold. Conjecture 3's strong form is thinner than Crocker's sequence (the Pell numbers vary as (1 + sqrt(2))^n, with 1 + sqrt(2) = 2.414... > 2) and so a failure wouldn't be surprising. But the weak form allows negative numbers, so it seems almost sure to hold. Does anyone have thoughts on these, especially the weak form of Conjecture 3? Breaking it into classes: 1. Sum of an odd prime and two positive Pell numbers  the strong form has only this form. 2. Sum of an odd prime, and the positive sum of two Pell numbers, one positive and one nonpositive. 3. Sum of an odd prime, and the nonpositive sum of two Pell numbers. Given a number n, it's easy to check if it has a form in class 1 or 2. But how do you rule out the possibility that it's in class 3? Choosing arbitrarily large (in absolute value) negative Pell numbers is allowed, so how could a purported counterexample be checked? Further, there should be something like O((log n)^2) numbers between n and 0 that are the sum of two Pell numbers, so the 'chance' that a number will have the form of such a negative number plus a prime is large, as log^2 n * n/log n >> n. 
20090206, 20:49  #8 
"Richard B. Woods"
Aug 2002
Wisconsin USA
7692_{10} Posts 
Sun now has a website for his stuff:
"Mixed Sums of Primes and Other Terms" http://math.nju.edu.cn/~zwsun/MSPT.htm established on Groundhog Day, appropriately enough (for those who've seen the movie). (and I was primenumbered visitor "00000353"  Is there a visitorcounter that doesn't display leading zeros?) Last fiddled with by cheesehead on 20090206 at 20:50 
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