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An identity Ramanujan probably missed

Susil Kumar Jena (2015)

Colloquium Mathematicae

"Ramanujan's 6-10-8 identity" inspired Hirschhorn to formulate his "3-7-5 identity". Now, we give a new "6-14-10 identity" which we suppose Ramanujan would have discovered but missed to mention in his notebooks.

Complete solutions of a Lebesgue-Ramanujan-Nagell type equation

Priyanka Baruah, Anup Das, Azizul Hoque (2024)

Archivum Mathematicum

We consider the Lebesgue-Ramanujan-Nagell type equation x 2 + 5 a 13 b 17 c = 2 m y n , where a , b , c , m 0 , n 3 and x , y 1 are unknown integers with gcd ( x , y ) = 1 . We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all S -integral points on a class of elliptic curves.

Congruent numbers with higher exponents

Florian Luca, László Szalay (2006)

Acta Mathematica Universitatis Ostraviensis

This paper investigates the system of equations x 2 + a y m = z 1 2 , x 2 - a y m = z 2 2 in positive integers x , y , z 1 , z 2 , where a and m are positive integers with m 3 . In case of m = 2 we would obtain the classical problem of congruent numbers. We provide a procedure to solve the simultaneous equations above for a class of the coefficient a with the condition gcd ( x , z 1 ) = gcd ( x , z 2 ) = gcd ( z 1 , z 2 ) = 1 . Further, under same condition, we even prove a finiteness theorem for arbitrary nonzero a .

Currently displaying 21 – 40 of 330