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On Alternatives of Polynomial Congruences

Mariusz Skałba (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

What should be assumed about the integral polynomials f ( x ) , . . . , f k ( x ) in order that the solvability of the congruence f ( x ) f ( x ) f k ( x ) 0 ( m o d p ) for sufficiently large primes p implies the solvability of the equation f ( x ) f ( x ) f k ( x ) = 0 in integers x? We provide some explicit characterizations for the cases when f j ( x ) are binomials or have cyclic splitting fields.

On an additive problem of unlike powers in short intervals

Qingqing Zhang (2022)

Czechoslovak Mathematical Journal

We prove that almost all positive even integers n can be represented as p 2 2 + p 3 3 + p 4 4 + p 5 5 with | p k k - 1 4 N | N 1 - 1 / 54 + ε for 2 k 5 . As a consequence, we show that each sufficiently large odd integer N can be written as p 1 + p 2 2 + p 3 3 + p 4 4 + p 5 5 with | p k k - 1 5 N | N 1 - 1 / 54 + ε for 1 k 5 .

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