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2008-12-23, 22:29   #1

"Richard B. Woods"
Aug 2002
Wisconsin USA

22·3·641 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:
 A surprising conjecture: n=x^2+T_y+F_m Sunday, December 21, 2008 12:28 PM From: "Zhi-Wei SUN" Dear number theorists, Recall that triangular numbers have the form T_n=n(n+1)/2 (n=0,1,2,...), and the Fibonacci numbers are given by F_0=0, F_1=1, and F_{n+1}=F_n+F_{n-1} (n=1,2,3,...). Here I pose the following somewhat suprising conjecture: Conjecture (Z. W. Sun, 2008) Each natural number can be written as the sum of a square, a triangular number and a Fibonacci number. Furthermore, c= lim inf_n r(n)/log n is a positive constant, where r(n) denotes the number of ways to write n in the form x^2+T_y+F_m with x,y,m non-negative integers. I have verified the conjecture for all natural numbers not exceeding 2,300,000. By my computation, c should be smaller than 2 and probably 0
"all natural numbers not exceeding 2,300,000" -- pretty good, but the next day brought:

Quote:
 From: "Zhi-Wei SUN" Dear all, Though I verified my incredible conjecture on n=x^2+T_y+F_m for n not more than 2,300,000, Dr. Douglas McNeil from University of London has found counter-examples via an efficient program: The smallest counter-example is n= 3,970,902, another large counter-example is 776,218,485. I also proposed a variant of my conjecture: n=T_x+T_y+F_m. Yet, McNeil found counter-examples for this variant too: The first two counter-examples are 2,892,173 and 10,991,572; large counterexamples include 96,799,987 and 901,415,393. I'm sorry that I raised a wrong conjecture to you. Perhaps the only positive thing is that it provides a very good example for R.K. Guy's Strong Law of Small Numbers. Merry Christmas! Zhi-Wei Sun - - - From: "Giovanni Resta" Zhi-Wei SUN wrote: > Dear number theorists, > > Recall that triangular numbers have the form T_n=n(n+1)/2 > (n=0,1,2,...), and the Fibonacci numbers are given by > > F_0=0, F_1=1, and F_{n+1}=F_n+F_{n-1} (n=1,2,3,...). > > Here I pose the following somewhat suprising conjecture: > > Conjecture (Z. W. Sun, 2008) Each natural number can be written as the > sum of a square, a triangular number and a Fibonacci number. > Furthermore, c= lim inf_n r(n)/log n is a positive constant, where > r(n) denotes the number of ways to write n in the form x^2+T_y+F_m > with x,y,m non-negative integers. It seems to me that the number 3,970,902 is the fist one which cannot be expressed in that form. I've checked only up to 20,000,000 without finding any other such number (I've had just a couple of hours because I'm leaving for the holidays right now.) Giovanni Resta - - - From: "Kurt Foster" Based on Zhi-Wei Sun's statement in the original post, that > It's not very clear why the conjecture may hold. I decided to try to push the numerical search a bit further to see if a counterexample (non-representable n) turned up. The following considerations indicated a possibly practical approach: n = x^2 + y*(y+1)/2 + F_m if and only if 8*n+1 = 8*x^2 + (2*y+1)^2 + 8*F_m, if and only if 8*n + 1 - 8*F_m = 8*x^2 + (2*y+1)^2 for some m >= 0. Now if M == 1 (mod 8), (e.g. M = 8*n + 1 - 8*F_m), then M = 8*x^2 + (2*y+1)^2 if and only if the exact prime-power exponent of each prime p == 5 or 7 (mod 8) which divides M is even. I wrote a Pari-GP script to implement these criteria, and let it crunch away. Of course, it's entirely possible my script is writ wrong, but up to the search limit 40000000 (4 x 10^7), it produced two candidate non-representable n, n = 3970902 and n = 39022919 - - - From: "Bjorn Poonen" Dear Zhi-Wei Sun: Since the number of integers up to B of the form x^2+T_y is about a constant times B/sqrt(log B), heuristics predict that there should be infinitely many counterexamples to your conjecture. If I'm not mistaken, the first one is 3970902. Bjorn Poonen

Last fiddled with by cheesehead on 2008-12-23 at 22:34

 2008-12-23, 22:54 #2 CRGreathouse     Aug 2006 3×1,993 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.
 2008-12-24, 08:13 #3 R. Gerbicz     "Robert Gerbicz" Oct 2005 Hungary 1,531 Posts All counter example for the original conjecture up to 2^33 are: Code: Counter-example=3970902 Counter-example=39022919 Counter-example=102132857 Counter-example=110468517 Counter-example=368495972 Counter-example=391099413 Counter-example=395147912 Counter-example=421129348 Counter-example=452808398 Counter-example=776218485 Counter-example=1005771844 Counter-example=1485470432 Counter-example=3038310485 Counter-example=3263773338 Counter-example=3485976107 Counter-example=3640901241 Counter-example=3758331463 Counter-example=3784200441 Counter-example=3944795435 Counter-example=4014507719 Counter-example=4277741986 Counter-example=4397438442 Counter-example=4542739955 Counter-example=4757066466 Counter-example=5167438708 Counter-example=7130095749 Counter-example=7213669167 Counter-example=7424527675 Counter-example=7696559526 Counter-example=8309766941 Counter-example=8462583631
2008-12-26, 18:38   #4

"Richard B. Woods"
Aug 2002
Wisconsin USA

22×3×641 Posts

Quote:
 Originally Posted by CRGreathouse Well, so much for sums of thin sequences representing all integers!
Not so fast ...

Zhi-Wei Sun isn't easily discouraged.

From the NMBRTHRY mailing list comes:

Quote:
 A promising conjecture: n=p+F_s+F_t Tuesday, December 23, 2008 7:57 AM From: "Zhi-Wei SUN" To: undisclosed-recipients Dear number theorists, Let {F_n} be the Fibonacci sequence given by F_0=0, F_1=1, and F_{n+1}=F_n+F_{n-1} (n=1,2,3,...). I have the following very promising conjecture. CONJECTURE [Sun, Dec. 23, 2008]. Each integer n>4 can be written as the sum of an odd prime and two positive Fibonacci numbers. I have verified the conjecture for n not exceeding 10^8. Note that 18484776 cannot be written as the sum of a prime p=1(mod 3) and two Fibonacci numbers, and 11095254 cannot be represented as the sum of a prime p=1(mod 4) and two Fibonacci numbers. Now I provide certain evidence to support the above conjecture. (1) The number of triples with p a prime not exceeding n and F_s,F_t2 is the sum of a prime and at most three powers of two.] However, below a given n>4 there are more Fibonacci numbers than powers of two. Also, Crocker's trick involves some very special properties of powers of two and it does not work for Fibonacci numbers. This time I'm very confident that the conjecture holds. Any comments are welcome! Merry Christmas! Zhi-Wei Sun http://math.nju.edu.cn/~zwsun
followed by:

Quote:
 A summary concerning my conjecture: n=p+F_s+F_t Thursday, December 25, 2008 12:22 AM From: "Zhi-Wei SUN" To: undisclosed-recipients Dear number theorists, Here I summarize the recent developments concerning my conjecture on sums of a prime and two Fibonacci numbers. In the evening of Dec. 23, 2008, I formulated the following stronger version of the conjecture and verified it up to 3*10^7. CONJECTURE. Any integer n>4 can be written as the sum of an odd prime, an ODD Fibonacci number and a positive Fibonacci number. Later I informed this to some authors of this mailing list including Prof. Douglas McNeil (London) and Bjorn Poonen (MIT). My former student Song Guo and Prof. McNeil have verified my above conjecture up to 4*10^8 and 2*10^9 respectively, and found NO counterexamples. McNeil told me that he would like to push the verification to 10^10. On Dec. 23, I could not find an integer 44 not representable as the sum of a prime p=5(mod 6) and two Fibonacci numbers; he found two such numbers below 2*10^9: 857530546 and 278749244. Sincerely, Zhi-Wei Sun http://math.nju.edu.cn/~zwsun

Quote:
 Originally Posted by CRGreathouse Kind of makes me wish I'd pushed ahead to find the counterexample myself.
The clock is running ...

Last fiddled with by cheesehead on 2008-12-26 at 18:43

2008-12-26, 19:22   #5
CRGreathouse

Aug 2006

3·1,993 Posts

Quote:
 Originally Posted by cheesehead The clock is running ...
I checked his earlier version of this conjecture (weaker, so I'm searching for a stronger counterexample) up to 2.6 x 10^10; I'm in progress for a verification to 6e10. Nothing yet.

2008-12-26, 20:23   #6
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

1,531 Posts

Quote:
 Originally Posted by cheesehead Not so fast ... Zhi-Wei Sun isn't easily discouraged. From the NMBRTHRY mailing list comes: followed by: The clock is running ...
It looks like he has got at least one cojecture/day.

2009-01-15, 20:08   #7
CRGreathouse

Aug 2006

597910 Posts

Quote:
 Originally Posted by R. Gerbicz It looks like he has got at least one cojecture/day.
I love when he (rarely) adds timestamps in addition to datastamps to his conjectures. Who else...

He's now offering Erdős-style prizes for proofs or counterexamples:
http://listserv.nodak.edu/cgi-bin/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.

 2009-02-06, 20:49 #8 cheesehead     "Richard B. Woods" Aug 2002 Wisconsin USA 22×3×641 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 prime-numbered visitor "00000353" -- Is there a visitor-counter that doesn't display leading zeros?) Last fiddled with by cheesehead on 2009-02-06 at 20:50

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