400 research outputs found
On the existence of -Diophantine quadruples
Let be a set of primes. We call an -tuple of
distinct, positive integers -Diophantine, if for all the integers
have only prime divisors coming from the set , i.e. if
all are -units. In this paper, we show that no -Diophantine
quadruple (i.e.~) exists if . Furthermore we show that for all
pairs of primes with and no
-Diophantine quadruples exist, provided that is not a
Wieferich prime pair
On Simultaneous Palindromes
A palindrome in base is an integer that remains the same when its
digit expansion in base is reversed. Let and be given distinct
integers . In this paper we discuss how many integers are palindromes in
base and simultaneously palindromes in base
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