A174885 Prime hypotenuses c with concatenation p = c||a||b a prime number, where (a,b,c) is a primitive Pythagorean triple with a the even leg.

29, 409, 461, 661, 929, 1249, 1289, 1381, 1801, 1901, 2081, 2609, 2621, 2749, 3041, 3301, 3881, 5309, 5701, 6421, 6481, 6521, 6529, 7349, 7489, 7789, 8641, 8849, 9349, 9629, 9649, 9689, 9829, 10321, 10709, 10861, 12841, 14321, 14561, 15061, 16661

Offset

1,1

Comments

See comments in A174825

c is the prime hypotenuse c, i. e. c of a primitive Pythagorean triple: a^2 + b^2 = c^2

References

W. W. R. Ball, H. S. M. Coxeter: Mathematical Recreations and Essays, New York: Dover, 1987

L. E. Dickson, "Rational Right Triangles", ch. 4 in History of the Theory of numbers, vol. II, Dover Publications 2005

W. Sierpinski, Pythagorean Triangles, Mineola, NY, Dover Publications, Inc, 2003

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(1) = 29 with 29^2 = 20^2 + 21^2 and 29 and 29||20||21 = 292021 prime.
a(2) = 409 with 409^2 = 120^2 + 391^2 and 409 and 409||120||391 = 409120391 prime.
a(3) = 461 with 461^2 = 380^2 + 261^2 and 461 and 461||380||261 = 461380261 prime.
a(4) = 661 with 661^2 = 300^2 + 589^2 and 661 and 661||300||589 = 661300589 prime.
a(5) = 929 with 929^2 = 920^2 + 129^2 and 929 and 929||920||129 = 929920129 prime.

Maple

N:= 20000: # for terms <= N
S:= {}:
for x from 2 while x^2 < N do
  for y from 3 - (x mod 2) to x-1 by 2 while x^2 + y^2 <= N do
    if igcd(x, y) > 1 then next fi;
    c:= x^2 + y^2;
    if not isprime(c) then next fi;
    a:= 2*x*y;
    b:= x^2-y^2;
    if isprime((c*10^(1+ilog10(a))+a)*10^(1+ilog10(b))+b) then
      S:= S union {c}
    fi
od od:
sort(convert(S, list)); # Robert Israel, Mar 06 2026

Crossrefs

Cf. A008846, A020882, A048161, A067756, A138044, A174825.

Sequence in context: A010945 A022624 A042630 * A264160 A210302 A163021

Adjacent sequences: A174882 A174883 A174884 * A174886 A174887 A174888

Keyword

base,nonn

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 01 2010

Extensions

More terms from Zak Seidov, Apr 04 2010

Edited by Robert Israel, Mar 07 2026

Status

approved