A154536 Positive integers that can be written as the sum of a positive Pell number and twice a positive Pell number.

3, 4, 5, 6, 7, 9, 11, 12, 14, 15, 16, 22, 25, 26, 29, 31, 33, 36, 39, 53, 59, 60, 63, 70, 72, 74, 80, 87, 94, 128, 141, 142, 145, 152, 169, 171, 173, 179, 193, 210, 227, 309, 339, 340, 343, 350, 367, 408, 410, 412

Offset: 1

Author

Zhi-Wei Sun, Jan 11 2009

Keywords

• nonn

Comments

On Jan 10 2009, Zhi-Wei Sun conjectured that any integer greater than 5 can be expressed as the sum of an odd prime and a term in the above sequence; in other words, each n=6,7,... can be written in the form p+P_s+2*P_t with p an odd prime and s,t>0. This has been verified up to 5*10^13 by D. S. McNeil (from London Univ.). Motivated by this conjecture, Qing-Hu Hou (from Nankai Univ.) observed and Zhi-Wei Sun proved that each term a(n) in the above sequence can be uniquely written in the form P_s+2P_t with s,t>0. Sun noted that 2176 cannot be written as the sum of a prime and two Pell numbers; D. S. McNeil found that 393185153350 cannot be written in the form p+P_s+3P_t and 872377759846 cannot be written in the form p+P_s+4P_t, where p is a prime and s and t are nonnegative.

Zhi-Wei Sun has offered a monetary reward for settling this conjecture.

Examples

			For n=12 the a(12)=22 solution is 22 = P_4 + 2*P_3.
		

References

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

Links

• Zhi-Wei SUN, Table of n, a(n), n=1..179.
• D. S. McNeil, Sun's strong conjecture
• D. S. McNeil, Various and sundry: a report on Sun's conjectures
• Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t
• Terence Tao, A remark on primality testing and decimal expansions, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
• 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

Crossrefs

Cf. A000129, A000040, A154257, A154285, A154364, A154417, A154421, A156695.

Programs

Mathematica

P[n_]:=P[n]=2*P[n-1]+P[n-2] P[0]=0 P[1]=1 i:=0 Do[Do[If[n==2*P[x]+P[y],i=i+1;Print[i," ",n]], {x,1,Max[1,Log[2,n]]},{y,1,Log[2,n]+1}]; Continue,{n,1,100000}]

Extensions

Mentioned McNeil's verification record for the representation n = p + P_s + 2P_t and his examples for n not of the form p + P_s + 3P_t and n not of the form p + P_s + 4P_t. - Zhi-Wei Sun, Jan 17 2009

D. S. McNeil has verified the conjecture up to 5*10^13. - Zhi-Wei Sun, Jan 20 2009