Page 1 Next

Displaying 1 – 20 of 103

Showing per page

A note on Sierpiński's problem related to triangular numbers

Maciej Ulas (2009)

Colloquium Mathematicae

We show that the system of equations t x + t y = t p , t y + t z = t q , t x + t z = t r , where t x = x ( x + 1 ) / 2 is a triangular number, has infinitely many solutions in integers. Moreover, we show that this system has a rational three-parameter solution. Using this result we show that the system t x + t y = t p , t y + t z = t q , t x + t z = t r , t x + t y + t z = t s has infinitely many rational two-parameter solutions.

A variety of Euler's sum of powers conjecture

Tianxin Cai, Yong Zhang (2021)

Czechoslovak Mathematical Journal

We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system n = a 1 + a 2 + + a s - 1 , a 1 a 2 a s - 1 ( a 1 + a 2 + + a s - 1 ) = b s has positive integer or rational solutions n , b , a i , i = 1 , 2 , , s - 1 , s 3 . Using the theory of elliptic curves, we prove that it has no positive integer solution for s = 3 , but there are infinitely many positive integers n such that it has a positive integer solution for s 4 . As a corollary, for s 4 and any positive integer n , the above Diophantine system has a positive rational solution. Meanwhile, we give conditions such that...

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.

Bihomogeneous forms in many variables

Damaris Schindler (2014)

Journal de Théorie des Nombres de Bordeaux

We count integer points on varieties given by bihomogeneous equations using the Hardy-Littlewood method. The main novelty lies in using the structure of bihomogeneous equations to obtain asymptotics in generically fewer variables than would be necessary in using the standard approach for homogeneous varieties. Also, we consider counting functions where not all the variables have to lie in intervals of the same size, which arises as a natural question in the setting of bihomogeneous varieties.

Cubic moments of Fourier coefficients and pairs of diagonal quartic forms

Jörg Brüdern, Trevor D. Wooley (2015)

Journal of the European Mathematical Society

We establish the non-singular Hasse principle for pairs of diagonal quartic equations in 22 or more variables. Our methods involve the estimation of a certain entangled two-dimensional 21st moment of quartic smooth Weyl sums via a novel cubic moment of Fourier coefficients.

Currently displaying 1 – 20 of 103

Page 1 Next