1,7

Zhi-Wei Sun conjectured that a(n)>0 for every n=5,6,...; in other words, any integer n>4 can be written as the sum of an odd prime, a positive Fibonacci number and a cube of a positive Fibonacci number, with one of two Fibonacci numbers odd. He has verified this up to 3*10^7.

Zhi-Wei Sun (zwsun(AT)nju.edu.cn) has offered a monetary reward for settling this conjecture.

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.

Z. W. Sun and M. H. Le, Integers not of the form c(2^a+2^b)+p^{alpha}, Acta Arith. 99(2001), 183-190.

K. J. Wu and Z. W. Sun, Covers of the integers with odd moduli and their applications to the forms x^m-2^n and x^2-F_{3n}/2, Math. Comp., in press. arXiv:math.NT/0702382

Zhi-Wei SUN, Table of n, a(n), n=1..50000.

Douglas McNeil, Sun's strong conjecture

Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t

Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t

For n=14 the a(14)=4 solutions are 3+F_4+(F_3)^3, 5+F_2+(F_3)^3, 5+F_6+(F_2)^3, 11+F_3+(F_2)^3

PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[(Mod[n, 2]==0||Mod[x, 3]>0)&&PQ[n-(Fibonacci[x])^3-Fibonacci[y]], 1, 0], {x, 2, 2*Log[2, n^(1/3)+1]}, {y, 2, 2*Log[2, Max[2, n-(Fibonacci[x])^3]]}] Do[Print[n, " ", RN[n]]; Continue, {n, 1, 50000}]

Cf. A154257, A154258, A000040, A000045

Sequence in context: A101872 A069929 A101312 this_sequence A035942 A036989 A035197

Adjacent sequences: A154260 A154261 A154262 this_sequence A154264 A154265 A154266

nonn

Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Jan 06 2009