A038198 Numbers n such that n^2 + 7 is a power of 2.

1, 3, 5, 11, 181

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• fini
• full

Comments

The exponents of the corresponding powers of 2 are 3, 4, 5, 7, 15 (see Ramanujan). - N. J. A. Sloane, Jun 01 2014

The terms lead to identities resembling Machin's Pi/4 = arctan(1/1) = 4*arctan(1/5) - arctan(1/239), for example, arctan(sqrt(7)/1) = 5*arctan(sqrt(7)/11) + 2*arctan(sqrt(7)/181), which can also be expressed as arcsin(sqrt(7/2^3)) = 5*arcsin(sqrt(7/2^7)) + 2*arcsin(sqrt(7/2^15)) (cf. A168229). - Joerg Arndt, Nov 09 2012

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 181, p. 56, Ellipses, Paris 2008.
• L. J. Mordell, Diophantine Equations, Academic Press, NY, 1969, p. 205.
• S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962. See Question 464, p. 327. - N. J. A. Sloane, Jun 01 2014

Links

• Spencer De Chenne, The Ramanujan-Nagell Theorem: Understanding the Proof
• Eric Weisstein's World of Mathematics, Ramanujan's Square Equation
• Wikipedia, Lucas Sequence

Crossrefs

Cf. A060728, A002249, A107920.

Programs

Mathematica

ok[n_] := Reduce[k>0 && n^2 + 7 == 2^k, k, Integers] =!= False; Select[Range[1000], ok] (* Jean-François Alcover, Sep 21 2011 *)

PARI

[x | n<-[0..99], issquare(2^n-7,&x)] \\ M. F. Hasler, Mar 11 2024