A075114 Perfect powers n such that 2n + 1 is a perfect power; the value of y^b in the solution of the Diophantine equation x^a - 2y^b = 1.

4, 121, 144, 4900, 166464, 5654884, 192099600, 6525731524, 221682772224, 7530688524100, 255821727047184, 8690408031080164, 295218051329678400, 10028723337177985444, 340681375412721826704

Offset: 1

Author

Zak Seidov, Oct 11 2002

Keywords

• more
• nonn

Comments

Note that the first ten numbers in this sequence are all squares. Except for 121, these squares are the y^2 in the Pell equation x^2 - 2y^2 = 1, whose solutions (x,y) are in sequences A001541 and A001542. The equation x^a - 2y^b = 1 is very similar to Catalan's equation x^a - y^b = 1, which has only one solution. Bennett shows that the equation x^2 - 2y^b = 1 has no solutions for b>2. Hence all the terms in this sequence are squares and solutions other than the Pell solutions must satisfy x^a - 2y^2 = 1 for a>2. The one known solution is 3^5 - 2*11^2 = 1. Are there any others? - T. D. Noe, Mar 29 2006

Links

• M. A. Bennett, Products of Consecutive Integers, Bull. London Math. Soc. 36 (2004), 683-694

Crossrefs

Cf. A001597.

Cf. A117547 (square root of terms).

Programs

Mathematica

pp = Select[ Range[10^8], Apply[ GCD, Last[ Transpose[ FactorInteger[ # ]]]] > 1 & ]; Select[pp, Apply[GCD, Last[ Transpose[ FactorInteger[2# + 1]]]] > 1 & ]
lim=10^14; lst={}; k=2; While[n=Floor[lim^(1/k)]; n>1, lst=Join[lst,Range[2,n]^k]; k++ ]; lst=Union[lst]; Intersection[lst,(lst-1)/2] (*T. D. Noe, Mar 29 2006 *)

Formula

Empirical G.f.: x*(117*x^4-4091*x^3+3951*x^2+19*x-4) / ((x-1)*(x^2-34*x+1)). - Colin Barker, Dec 21 2012

Extensions

Extended by Robert G. Wilson v, Oct 15 2002

More terms from T. D. Noe, Mar 29 2006

More terms from T. D. Noe, Nov 19 2006