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Hypotenuse numbers w of Pythagorean triples (u, v, w) for which (u^2, v^2, w^2) is an "abc-hit".
3

%I #26 Oct 19 2023 07:36:55

%S 25,41,65,125,145,289,337,377,425,625,677,841,845,1025,1201,1625,1681,

%T 1985,2125,2197,2305,2873,3125,3281,3425,3721,4097,4225,4481,4705,

%U 4825,4901,4913,5329,6401,6625,6725,6845,7585,7813,7817,8065,8177,9409,10625,10985

%N Hypotenuse numbers w of Pythagorean triples (u, v, w) for which (u^2, v^2, w^2) is an "abc-hit".

%C (a, b, c) is an ABC triple if gcd(a, b) = 1 and a + b = c. ABC triples with c > rad(a*b*c) are called "abc-hits". For primitive Pythagorean triples (u, v, w) it is u^2 + v^2 = w^2 and gcd(u^2, v^2) = 1. (u^2, v^2, w^2) are therefore ABC triples. They are then "abc-hits" if in addition w^2 > rad(u^2*v^2*w^2). If (u, v, w) is a non-primitive Pythagorean triple, (u^2, v^2, w^2) is not an ABC triple.

%C The corresponding values of min(u, v) and max(u, v) are in the sequences A366674 and A366675.

%C w of primitive Pythagorean triples (u, v, w) with A007947(u^2*v^2*w^2) < w^2.

%C Subsequence of intersection of A020882 and sqrt(A130510).

%H Felix Huber, <a href="/A366428/a366428.txt">Pythagorean triples (u, v, w) for which (u^2, v^2, w^2) is an "abc-hit"</a>

%H Abderrahmane Nitaj, <a href="https://nitaj.users.lmno.cnrs.fr/abc.html">The ABC Conjecture Home Page</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Abc_conjecture">abc conjecture</a>

%e 25 from the primitive Pythagorean triple (7, 24, 25) is in the sequence, because 7^2 + 24^2 = 25^2, gcd(7^2, 24^2) = 1 and 25^2 = 625 > rad(7^2*24^2*25^2) = 7*2*3*5 = 210.

%Y Cf. A366674, A366675 (corresponding values of min(u, v) and max(u, v)).

%Y Cf. A020882 (hypotenuses of primitive Pythagorean triangles), A130510 ("abc-hits"), A007947 (squarefree kernel).

%K nonn

%O 1,1

%A _Felix Huber_, Oct 13 2023