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A154536
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Positive integers that can be written as the sum of a positive Pell number and twice a positive Pell number.
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10
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
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LINKS
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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
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EXAMPLE
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For n=12 the a(12)=22 solution is 22 = P_4 + 2*P_3.
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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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
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STATUS
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approved
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