[seqfan] FYI - conjecture: n=x^2+T_y+F_m

Alexander Povolotsky apovolot at gmail.com
Mon Dec 22 15:51:03 CET 2008

FYI - Perhaps someone might be interested to generate the sequence,
based on the number of ways to write n in the form x^2+T_y+F_m
with x,y,m non-negative integers - per below.

From:Zhi-Wei SUN <zwsun at nju.edu.cn>
Subject : A surprising conjecture: n=x^2+T_y+F_m
Date : Sun, Dec 21, 2008 01:29 PM

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 surprising 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<c<1.
I have calculated r(n)/log n for n between 20,000,001 and 20,000,500;
most of the values are greater than 1 and smaller than 4.

   It's not very clear why the conjecture may hold. Note that
2^{m/2}-1\le F_m<2^m and |{y\ge 0: T_y\le n}|  is about sqrt{2n}. Also,
{T_y: y=0,1,2,...}
contains a complete system of residues modulo any power of two.
[Moreover,  I and my student W. Zhang [arXiv:0812.3089] recently
showed that for any given positive integer k, the set {binomial coeff.
C(n,k): n=0,1,2,...}  contains a complete system of residues modulo
any powers of two, i.e., it is a dense subset of the ring Z_2of 2-adic

  Compared with the representation function for Goldbach's conjecture,
and the representation function for my conjecture on sums of primes and
triangular numbers
[arXiv:0803.3737], the above representation function r(n) grows much
slower! It seems that the above conjecture is the first example for
representation of all natural
  numbers with the representation function growing so slow. Thus, in my
opinion, if the above conjecture holds, its proof would be very difficult.

  Any comments are welcome!


Zhi-Wei Sun (Nanjing University, China)

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