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} corresponds to {a(n), A067755(n), A067756(n)} - Lekraj Beedassy (blekraj(AT)yahoo.com), 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). [Sierpinski]

W. Sierpinski, 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, Journal of Integer Sequences, Vol. 4(2001), #01.2.3, Prime Pythagorean triangles

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.

Select[Prime[Range[200]], PrimeQ[(#^2 + 1)/2] &] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 07 2006

(PARI) a(n)=local(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

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

Sequence in context: A089439 A122516 A023233 this_sequence A051642 A007671 A090471

Adjacent sequences: A048158 A048159 A048160 this_sequence A048162 A048163 A048164

nonn,nice

Harvey Dubner (harvey(AT)dubner.com)

More terms from David W. Wilson (davidwwilson(AT)comcast.net)