On the prime factors of (²ⁿ_{𝑛})

Abstract

Several quantitative results are given expressing the fact that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis Subscript n Baseline Superscript 2 n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(_n^{2n})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is usually divisible by a high power of the small primes. On the other hand, it is shown that for any two primes <italic>p</italic> and <italic>q</italic>, there exist infinitely many <italic>n</italic> for which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis left-parenthesis Subscript n Baseline Superscript 2 n Baseline right-parenthesis comma p q right-parenthesis equals 1"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">((_n^{2n}),pq) = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Locations

• Mathematics of Computation - View - PDF