1,1

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

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.

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

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.

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 *)

(PARI) {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 */

(Haskell)

a048161 n = a048161_list !! (n-1)

a048161_list = [p | p <- a065091_list, a010051 ((p^2 + 1) `div` 2) == 1]

-- Reinhard Zumkeller, Aug 26 2012

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

(Python)

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

Cf. A067755, A067756. Complement in primes of A094516.

Cf. A017281, A154428, A010051, A065091, A005383.

Cf. A048270, A048295, A308635, A308636. Primes contained in A002731.

Sequence in context: A284036 A172438 A023233 * A284034 A051642 A007671

Adjacent sequences: A048158 A048159 A048160 * A048162 A048163 A048164

nonn,easy,nice

Harvey Dubner (harvey(AT)dubner.com)

More terms from David W. Wilson

approved