A308635 -id:A308635 - cp's OEIS frontend

A048161 Primes p such that q = (p^2 + 1)/2 is also a prime.

3, 5, 11, 19, 29, 59, 61, 71, 79, 101, 131, 139, 181, 199, 271, 349, 379, 409, 449, 461, 521, 569, 571, 631, 641, 661, 739, 751, 821, 881, 929, 991, 1031, 1039, 1051, 1069, 1091, 1129, 1151, 1171, 1181, 1361, 1439, 1459, 1489, 1499, 1531, 1709, 1741, 1811, 1831, 1901

Offset: 1

Harvey Dubner (harvey(AT)dubner.com)

Primes which are a leg of an integral right triangle whose hypotenuse is also prime.
It is conjectured that there are an infinite number of such triangles.
The Pythagorean triple {p, (p^2 - 1)/2, (p^2 + 1)/2} corresponds to {a(n), A067755(n), A067756(n)}. - Lekraj Beedassy, Oct 27 2003
There is no Pythagorean triangle all of whose sides are prime numbers. Still there are Pythagorean triangles of which the hypotenuse and one side are prime numbers, for example, the triangles (3,4,5), (11,60,61), (19,180,181), (61,1860,1861), (71,2520,2521), (79,3120,3121). [Sierpiński]
We can always write p=(Y+1)^2-Y^2, with Y=(p-1)/2, therefore q=(Y+1)^2+Y^2. - Vincenzo Librandi, Nov 19 2010
p^2 and p^2+1 are semiprimes; p^2 are squares in A070552 Numbers n such that n and n+1 are products of two primes. - Zak Seidov, Mar 21 2011

			For p=11, (p^2+1)/2=61; p=61, (p^2+1)/2=1861.
For p(1)=3, the right triangle 3, 4, 5 is the smallest where 5=(3*3+1)/2.
For p(10)=101, the right triangle is 101, 5100, 5101 where 5101=(101*101+1)/2.

Wacław Sierpiński, Pythagorean triangles, Dover Publications, Inc., Mineola, NY, 2003, p. 6. MR2002669

T. D. Noe, Table of n, a(n) for n = 1..10000
H. Dubner and T. Forbes, Prime Pythagorean triangles, Journal of Integer Sequences, Vol. 4(2001), #01.2.3.

Cf. A067755, A067756. Complement in primes of A094516.
Cf. A017281, A154428, A010051, A065091, A005383.
Cf. A048270, A048295, A308635, A308636. Primes contained in A002731.

a048161 n = a048161_list !! (n-1)
a048161_list = [p | p <- a065091_list, a010051 ((p^2 + 1) `div` 2) == 1]
-- Reinhard Zumkeller, Aug 26 2012

[p: p in PrimesInInterval(3, 2000) | IsPrime((p^2+1) div 2)]; // Vincenzo Librandi, Dec 31 2013

a := proc (n) if isprime(n) = true and type((1/2)*n^2+1/2, integer) = true and isprime((1/2)*n^2+1/2) = true then n else end if end proc: seq(a(n), n = 1 .. 2000) # Emeric Deutsch, Jan 18 2009

Select[Prime[Range[200]], PrimeQ[(#^2 + 1)/2] &] (* Stefan Steinerberger, Apr 07 2006 *)
a[ n_] := Module[{p}, If[ n < 1, 0, p = a[n - 1]; While[ (p = NextPrime[p]) > 0, If[ PrimeQ[(p*p + 1)/2], Break[]]]; p]]; (* Michael Somos, Nov 24 2018 *)

{a(n) = my(p); if( n<1, 0, p = a(n-1) + (n==1); while(p = nextprime(p+2), if( isprime((p*p+1)/2), break)); p)}; /* Michael Somos, Mar 03 2004 */

from sympy import isprime, nextprime; p = 2
while p < 1901: p = nextprime(p); print(p, end = ', ') if isprime((p*p+1)//2) else None # Ya-Ping Lu, Apr 24 2025

A000035(a(n))*A010051(a(n))*A010051((a(n)^2+1)/2) = 1. - Reinhard Zumkeller, Aug 26 2012

More terms from David W. Wilson

A048270 Sequence of 2 Pythagorean triangles, each with a leg and hypotenuse prime. The leg of the second triangle is the hypotenuse of the first.

Original entry on oeis.org

3, 11, 19, 59, 271, 349, 521, 929, 1031, 1051, 1171, 2381, 2671, 2711, 2719, 3001, 3499, 3691, 4349, 4691, 4801, 4999, 5591, 5669, 6101, 6359, 6361, 7159, 7211, 7489, 8231, 8431, 8761, 9241, 10099, 10139, 11719, 11821, 12239, 12281, 12781

Offset: 1

Harvey Dubner (harvey(AT)dubner.com)

nonn

It is conjectured that there are infinitely many such pairs of triangles.

Subsequence of A048161. - Lekraj Beedassy, Sep 16 2005

			p(1)=3 because 3 is prime, 5 = (3*3 + 1)/2 and 13 = (5*5 + 1)/2, 5, 13 both prime.

Ray Chandler, Table of n, a(n) for n = 1..10000
H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3.

Cf. A048161, A048295, A308635, A308636. Primes in A116945.

For each p(n), there is a q=(p*p+1)/2 and r=(q*q+1)/2 such that p, q, r are all prime.

More terms from Ray Chandler, Jun 12 2019

A048295 Sequence of 3 Pythagorean triangles, each with a leg and hypotenuse prime. The hypotenuse of each triangle is the leg of the next triangle.

Original entry on oeis.org

271, 349, 3001, 10099, 11719, 12281, 25889, 39901, 46399, 63659, 169219, 250361, 264169, 287629, 289049, 312581, 353081, 440681, 473009, 502501, 502961, 541951, 594751, 620491, 627911, 632699, 704581, 757111, 762899, 922261, 959269

Offset: 1

Harvey Dubner (harvey(AT)dubner.com)

nonn

			p(5)=271, q=36721, r=674215921, s=227283554064939121.

Ray Chandler, Table of n, a(n) for n = 1..10000
H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3.

Cf. A048161, A048270, A308635, A308636. Primes in A188546.

For each p(n), q=(p*p+1)/2, r=(q*q+1)/2, s=(r*r+1)/2 and p, q, r, s are all prime.

More terms from Ray Chandler, Jun 12 2019

A308636 Sequence of 5 Pythagorean triangles, each with a leg and hypotenuse prime. The hypotenuse of each triangle is the leg of the next triangle.

Original entry on oeis.org

356498179, 432448789, 5380300469, 10667785241, 11238777509, 12129977791, 23439934621, 28055887949, 33990398249, 34250028521, 34418992099, 34773959159, 34821663421, 36624331189, 40410959231, 43538725229, 47426774869

Offset: 1

Ray Chandler, Jun 12 2019

nonn

			p(1)=356498179, q=63545475815158021, r=63545475815158021, s=2038208257886801569993754841378314277932542447949256249537232302421, ...

Ray Chandler, Table of n, a(n) for n = 1..38
H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3.

Cf. A048161, A048270, A048295, A308635. Primes in A187431.

For each p(n), q=(p*p+1)/2, r=(q*q+1)/2, s=(r*r+1)/2, t=(s*s+1)/2, u=(t*t+1)/2 and p, q, r, s, t, u are all prime.

A105318 Starting prime for the smallest prime Pythagorean sequence for n triangles.

Original entry on oeis.org

5, 3, 271, 169219, 356498179, 2500282512131, 20594058719087111, 2185103796349763249

Offset: 1

Lekraj Beedassy, Apr 26 2005

Smallest prime p(0) such that the n-chain governed by recurrence p(i+1)=(p(i)^2 + 1)/2 are all primes. Equivalently, least prime p(0) that generates a sequence of n 2-prime triangles, where p(k) is the hypotenuse of the k-th triangle and the leg of the (k+1)-th triangle.

For n>2, the last digit of a(n) is 1 or 9. - Ya-Ping Lu, May 17 2025

			5 is a(1) because (5^2+1)/2 = 13 is prime, but (13^2+1)/2 = 85 is not.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 258.

C. K. Caldwell, The Prime Glossary, Pythagorean triples
H. Dubner, Posting to Number Theory List
H. Dubner and T. Forbes, Prime Pythagorean Triangles
H. Dubner and T. Forbes, Journal of Integer Sequences, Vol. 4(2001) #01.2.3, Prime Pythagorean triangles
T. Forbes, Posting to Number Theory List
T. Forbes, Posting to Number Theory List

Cf. A048161, A048270, A048295, A308635, A308636.

from sympy import isprime, nextprime; m = lambda x: (x*x+1)//2; p = 2; D = {}
while p < 2185103796349763249:
    p = nextprime(p); q = m(p); n = 1
    while isprime(q) and isprime(m(q)): n += 1; q = m(q)
    if n not in D: D.update({n: p})
[print(k, end =', ') for key, k in sorted(D.items())] # Ya-Ping Lu, May 17 2025

a(1) added by T. D. Noe, Jan 29 2011

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A048161 Primes p such that q = (p^2 + 1)/2 is also a prime.

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A048270 Sequence of 2 Pythagorean triangles, each with a leg and hypotenuse prime. The leg of the second triangle is the hypotenuse of the first.

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A048295 Sequence of 3 Pythagorean triangles, each with a leg and hypotenuse prime. The hypotenuse of each triangle is the leg of the next triangle.

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A308636 Sequence of 5 Pythagorean triangles, each with a leg and hypotenuse prime. The hypotenuse of each triangle is the leg of the next triangle.

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A105318 Starting prime for the smallest prime Pythagorean sequence for n triangles.

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