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• W4312082034 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p left-parenthesis n right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>p(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p left-parenthesis upper Q cubed script l n plus beta right-parenthesis identical-to 0 left-parenthesis mod script l right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>Q</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mi>β<!-- β --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≡<!-- ≡ --></mml:mo> <mml:mn>0</mml:mn> <mml:mspace width=0.667em /> <mml:mo stretchy=false>(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width=0.333em /> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>p( Q^3 ell n+beta )equiv 0pmod ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script l> <mml:semantics> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:annotation encoding=application/x-tex>ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Q> <mml:semantics> <mml:mi>Q</mml:mi> <mml:annotation encoding=application/x-tex>Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are prime and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=5 less-than-or-equal-to script l less-than-or-equal-to 31> <mml:semantics> <mml:mrow> <mml:mn>5</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>31</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>5leq ell leq 31</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; these lie in two natural families distinguished by the square class of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1 minus 24 beta left-parenthesis mod script l right-parenthesis> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mn>24</mml:mn> <mml:mi>β<!-- β --></mml:mi> <mml:mspace width=0.667em /> <mml:mo stretchy=false>(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width=0.333em /> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>1-24beta pmod ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In recent decades much work has been done to understand congruences of the form <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p left-parenthesis upper Q Superscript m Baseline script l n plus beta right-parenthesis identical-to 0 left-parenthesis mod script l right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>Q</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mi>β<!-- β --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≡<!-- ≡ --></mml:mo> <mml:mn>0</mml:mn> <mml:mspace width=0.667em /> <mml:mo stretchy=false>(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width=0.333em /> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>p(Q^mell n+beta )equiv 0pmod ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is now known that there are many such congruences when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m greater-than-or-equal-to 4> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>mgeq 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, that such congruences are scarce (if they exist at all) when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m equals 1 comma 2> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>m=1, 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and that for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m equals 0> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>m=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such congruences exist only when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script l equals 5 comma 7 comma 11> <mml:semantics> <mml:mrow> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> <mml:mn>7</mml:mn> <mml:mo>,</mml:mo> <mml:mn>11</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>ell =5, 7, 11</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For congruences like Atkin’s (when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m equals 3> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>m=3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>), more examples have been found for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=5 less-than-or-equal-to script l less-than-or-equal-to 31> <mml:semantics> <mml:mrow> <mml:mn>5</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>31</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>5leq ell leq 31</mml:annotation> </mml:semantics> </mml:math> </inline-formula> but little else seems to be known. Here we use the theory of modular Galois representations to prove that for every prime <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script l greater-than-or-equal-to 5> <mml:semantics> <mml:mrow> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>ell geq 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there are infinitely many congruences like Atkin’s in the first natural family which he discovered and that for at least <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=17 slash 24> <mml:semantics> <mml:mrow> <mml:mn>17</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>24</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>17/24</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the primes <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script l> <mml:semantics> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:annotation encoding=application/x-tex>ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there are infinitely many congruences in the second family." @default.
• W4312082034 created "2023-01-04" @default.
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• W4312082034 date "2022-12-08" @default.
• W4312082034 modified "2023-09-26" @default.
• W4312082034 title "Congruences like Atkin’s for the partition function" @default.
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