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- W3127260892 abstract "Abstract We investigate the function $$P_{a,b;N}(ell )$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>;</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> describing the number of $$ell $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>ℓ</mml:mi> </mml:math> -th powers among the first N terms of an arithmetic progression $$ax+b$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> </mml:math> . We completely describe the arithmetic progressions containing the most $$ell $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>ℓ</mml:mi> </mml:math> -th powers asymptotically. Based on these results we formulate problems concerning the maximum of $$P_{a,b;N}(ell )$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>;</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , and we give affirmative answers to these questions for certain small values of $$ell $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>ℓ</mml:mi> </mml:math> and N ." @default.
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- W3127260892 date "2021-01-30" @default.
- W3127260892 modified "2023-10-17" @default.
- W3127260892 title "Powers in arithmetic progressions" @default.
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- W3127260892 doi "https://doi.org/10.1007/s11139-020-00331-5" @default.
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