A160054 Primes prime(k) such that prime(k)^2 + prime(k+1)^2 - 1 is a perfect square.

7, 11, 23, 109, 211, 307, 1021, 4583, 42967, 297779, 1022443, 1459811, 10781809, 125211211, 11673806759, 3019843939831, 40047392632801, 88212019638251209, 444190204424015227, 57852556614292865039, 9801250757169593701501, 64747502900142088755541, 619216322498658374863033

Offset

1,1

Comments

An infinite number of solutions exists for a^2 + b^2 - 1 = c^2 over the set of natural numbers a, b, c.

If we constrain these to b=a+2, i.e., 2a^2 + 4a + 3 = c^2, the solutions are with a = 1, 11, 69, 407, 2377, ... (The twin prime 11 is also in this sequence here. The solutions can be generated recursively from a(0)=1, m(0)=3 and a(k+1) = 3*a(k) + 2*m(k) + 2, m(k+1) = 4*a(k) + 3*m(k) + 4.)

Filtering these solutions for prime pairs a(k) and b(k) would generate the subset of lower twin primes in the sequence.

The equivalent procedure can be carried out for other prime gaps 2*d such that prime(k)=a, prime(k+1)=a+2*d, 2*a^2 + 4*a*d + 4*d^2 - 1 = m^2. This decomposes the sequence into classes according to the gap 2*d.

a(17) > 5*10^12. - Donovan Johnson, May 17 2010

Links

Table of n, a(n) for n=1..23.

Formula

{A000040(k): A069484(k)-1 in A000290}.

Example

7^2 + 11^2 - 1 = 13^2.
11^2 + 13^2 - 1 = 17^2.
23^2 + 29^2 - 1 = 37^2.
109^2 + 113^2 - 1 = 157^2.
211^2 + 223^2 - 1 = 307^2.
307^2 + 311^2 - 1 = 19^2*23^2.
1021^2 + 1031^2 - 1 = 1451^2.
4583^2 + 4591^2 - 1 = 13^2*499^2.

Mathematica

lst = {}; p = q = 2; While[p < 4000000000, q = NextPrime@ p; If[ IntegerQ[ Sqrt[p^2 + q^2 - 1]], AppendTo[lst, p]; Print@ p]; p = q]; lst (* Robert G. Wilson v, May 31 2009 *)

Prog

(PARI) p=2; forprime(q=3, 1e6, if(issquare(q^2+p^2-1), print1(p", ")); p=q) \\ Charles R Greathouse IV, Nov 06 2014
(PARI) is(n)=issquare(n^2+nextprime(n+1)^2-1)&&isprime(n) \\ Charles R Greathouse IV, Nov 29 2014
(Magma) [n: n in [0..2*10^7] | IsSquare(n^2+NextPrime(n+1)^2-1) and IsPrime(n)]; // Vincenzo Librandi, Aug 02 2015

Crossrefs

Cf. A050791, A129288, A271050.

Sequence in context: A243916 A181841 A076855 * A359414 A247590 A027830

Adjacent sequences: A160051 A160052 A160053 * A160055 A160056 A160057

Keyword

nonn

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 01 2009

Extensions

Edited and 4 more terms from R. J. Mathar, May 08 2009

a(13) from Robert G. Wilson v, May 31 2009

a(15)-a(16) from Donovan Johnson, May 17 2010

More terms from Jinyuan Wang, Jan 09 2021

Status

approved