Powers in arithmetic progressions

Abstract

AbstractWe investigate the function $$P_{a,b;N}(\ell )$$ P a , b ; N ( ℓ ) describing the number of $$\ell $$ ℓ -th powers among the first N terms of an arithmetic progression $$ax+b$$ a x + b . We completely describe the arithmetic progressions containing the most $$\ell $$ ℓ -th powers asymptotically. Based on these results we formulate problems concerning the maximum of $$P_{a,b;N}(\ell )$$ P a , b ; N ( ℓ ) , and we give affirmative answers to these questions for certain small values of $$\ell $$ ℓ and N.