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A370763 Table read by rows: row n is the unique primitive Pythagorean triple (a,b,c) such that a = prime(n)*prime(n+1) and its long leg and hypotenuse are consecutive natural numbers.

%I A370763 #51 Mar 26 2024 05:32:52
%S A370763 15,112,113,35,612,613,77,2964,2965,143,10224,10225,221,24420,24421,
%T A370763 323,52164,52165,437,95484,95485,667,222444,222445,899,404100,404101,
%U A370763 1147,657804,657805,1517,1150644,1150645,1763,1554084,1554085,2021,2042220,2042221,2491,3102540,3102541
%N A370763 Table read by rows: row n is the unique primitive Pythagorean triple (a,b,c) such that a = prime(n)*prime(n+1) and its long leg and hypotenuse are consecutive natural numbers.
%C A370763 The pair of natural numbers (d,e) is said to be a pair of primitive twin divisors of a natural number m when d*e = m and gcd(d,e) = 1.
%C A370763 Given two prime numbers p and q (p<q), as the only pairs of primitive twin divisors of (p*q)^2 are (1,p^2*q^2) and (p^2,q^2), then, the only primitive Pythagorean triples with a short leg equal to p*q are (p*q,(p^2*q^2-1)/2,(p^2*q^2+1)/2) and (p*q,(q^2-p^2)/2,(q^2+p^2)/2), being (p*q,(p^2*q^2-1)/2,(p^2*q^2+1)/2) the only one that has consecutive long leg and hypotenuse.
%D A370763 Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
%H A370763 Miguel-Ángel Pérez García-Ortega, <a href="/A370763/a370763.pdf">Capítulo 5</a>.
%F A370763 Row n = (a, b, c) = (p * q, (p^2 * q^2 - 1)/2, (p^2 * q^2 + 1)/2), where p = prime(n) and q = prime(n+1).
%e A370763 Table begins:
%e A370763   n=2:   15,   112,   113;
%e A370763   n=3:   35,   612,   613;
%e A370763   n=4:   77,  2964,  2965;
%e A370763   n=5:  143, 10224, 10225;
%e A370763   n=6:  221, 24420, 24421;
%e A370763   ...
%t A370763 Apply[Join,Map[{#,(#^2-1)/2,(#^2+1)/2} &,Prime[Range[2,31]]Prime[Range[3, 32]]]]
%Y A370763 Cf. A000040, A006094 (short leg), A102770 (inradius).
%K A370763 nonn,easy,tabf
%O A370763 2,1
%A A370763 _Miguel-Ángel Pérez García-Ortega_, Mar 01 2024