1,6

Zhi-Wei Sun conjectured that a(n)>0 for all n=5,6,...; in other words, any integer n>4 can be represented as the sum of an odd prime, an odd Lucas number and a Lucas number. This has been verified up to 1.5*10^8. Sun thought that the constant lim inf_n a(n)/log(n) is greater than 2 and smaller than 3. For k=2,3 Sun also conjectured that any integer n>4 can be written in the form p+L_s+(L_t)^k, where p is an odd prime and L_s or L_t is odd.

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 summary concerning my conjecture n=p+F_s+F_t

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

For n=3 the a(7)=4 solutions are 3+L_1+L_2, 3+L_2+L_1, 5+L_1+L_1.

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

A000032, A000040, A154257, A154258, A154263

Sequence in context: A099900 A138728 A082332 this_sequence A036552 A132169 A112975

Adjacent sequences: A154282 A154283 A154284 this_sequence A154286 A154287 A154288

nice,nonn

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