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A recursive definition of p -ary addition without carry

François Laubie (1999)

Journal de théorie des nombres de Bordeaux

Let p be a prime number. In this paper we prove that the addition in p -ary without carry admits a recursive definition like in the already known cases p = 2 and p = 3 .

A remark on a Diophantine equation of S. S. Pillai

Azizul Hoque (2024)

Czechoslovak Mathematical Journal

S. S. Pillai proved that for a fixed positive integer a , the exponential Diophantine equation x y - y x = a , min ( x , y ) > 1 , has only finitely many solutions in integers x and y . We prove that when a is of the form 2 z 2 , the above equation has no solution in integers x and y with gcd ( x , y ) = 1 .

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