A051892 -id:A051892 - cp's OEIS frontend

A068501 Values m such that the consecutive pair parameters(m,m+1) generate Pythagorean triples whose odd terms are both prime.

1, 2, 5, 9, 14, 29, 30, 35, 39, 50, 65, 69, 90, 99, 135, 174, 189, 204, 224, 230, 260, 284, 285, 315, 320, 330, 369, 375, 410, 440, 464, 495, 515, 519, 525, 534, 545, 564, 575, 585, 590, 680, 719, 729, 744, 749, 765, 854, 870, 905, 915, 950, 974, 1080, 1119

Offset: 1

Lekraj Beedassy, Mar 25 2002

nonn

Setting u=m; v=m+1, triples (a,b,c) with a=u+v, b=2*u*v, c = u^2+v^2 = (a^2+1)/2 correspond to (A048161, A067755, A067756), a and c being both prime.

Zak Seidov, Table of n, a(n) for n = 1..10000
Robert Simms, Deriving Pythagorean Triples (web archive)

Cf. A051892.

lst={};Do[If[PrimeQ[(n+1)^2-n^2]&&PrimeQ[(n+1)^2+n^2],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 01 2010 *)
Reap[Do[a=Prime[k];If[PrimeQ[(a^2+1)/2],Sow[(a-1)/2]],{k,2,10^5}]][[2,1]](* Zak Seidov, Apr 16 2011 *)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 19 2002

A051859 Values of C (the hypotenuse) of a Pythagorean triangle with A (the short leg) and C both prime and part of a twin prime.

Original entry on oeis.org

5, 13, 61, 181, 421, 5101, 60901, 135721, 161881, 163021, 218461, 595141, 1108561, 2574181, 2740141, 3248701, 3535141, 3723721, 3729181, 8197201, 13933921, 20218441, 23605321, 28569241, 33874681, 47248921, 68667481, 69372421

Offset: 1

Stuart M. Ellerstein (ellerstein(AT)aol.com), Dec 14 1999

nonn

All terms of the sequence must be part of a Pythagorean triple of the form (2u-1), 2u*(u-1), (2u^2 - 2u + 1). - Joshua Zucker, May 12 2006

Harvey P. Dale, Table of n, a(n) for n = 1..1000

See A051642 for the A's and A051858 for the B's.

Subset of A067756.

tppQ[{a_,b_,c_}]:=AllTrue[{a,c},PrimeQ]&&AnyTrue[a+{2,-2},PrimeQ] && AnyTrue[ c+{2,-2},PrimeQ]; Select[Table[{2n-1,2n(n-1),2n^2-2n+1},{n,2,10000}],tppQ][[All,3]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 27 2021 *)

A051858 + 1, or 2*A051891^2 - 2*A051891 + 1, or 2*A051892^2 + 2*A051892 + 1. - Joshua Zucker, May 12 2006

More terms from Joshua Zucker, May 12 2006

A051891 Values of m, the main key or generating number for Pythagorean triangles in which S (the odd short leg) and U (the hypotenuse) are twin primes.

Original entry on oeis.org

2, 3, 6, 10, 15, 51, 175, 261, 285, 286, 331, 546, 745, 1135, 1171, 1275, 1330, 1365, 1366, 2025, 2640, 3180, 3436, 3780, 4116, 4861, 5860, 5890, 6036, 6670, 8646, 8961, 8980, 9715, 9945, 10116, 10180, 15771, 16590, 16876, 19326, 20215, 20266, 21451

Offset: 1

Stuart M. Ellerstein (ellerstein(AT)aol.com), Dec 17 1999

nonn

Robert Simms, Deriving Pythagorean Triples (web archive)

Cf. A051892. See A051642 for the S's, A051859 for the U's and A051858 for the T's (the even long leg).

a(n) = (A051642(n) + 1) / 2. - Sean A. Irvine, Oct 12 2021

More terms from Joshua Zucker, May 12 2006

cp's OEIS Frontend

A068501 Values m such that the consecutive pair parameters(m,m+1) generate Pythagorean triples whose odd terms are both prime.

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A051859 Values of C (the hypotenuse) of a Pythagorean triangle with A (the short leg) and C both prime and part of a twin prime.

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Mathematica

Formula

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A051891 Values of m, the main key or generating number for Pythagorean triangles in which S (the odd short leg) and U (the hypotenuse) are twin primes.

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Author

Keywords

Links

Crossrefs

Formula

Extensions