Sárközy's Theorem

This article deals with Sárközy's theorem on the freedom from squares of binomial mean coefficients. For the theorem about the existence of a perfect square in the difference set of a sequence of natural numbers, which is also often referred to as Sárközy's theorem, see Fürstenberg-Sárközy's theorem .

The set of Sárközy is part proof of Erdős's Freedom Square guess . This says that the mean binomial coefficient

${\displaystyle {2n \choose n}}$

for is never square-free . András Sárközy proved that one exists, so this is true for all of what is known as the Sárközy Theorem. He also showed in 1985 that ${\displaystyle n>4}$ ${\displaystyle n_{0}}$ ${\displaystyle n>n_{0}}$

${\displaystyle \ln s(n)\approx \left({\sqrt {2}}-2\right)\zeta \left({\frac {1}{2}}\right){\sqrt {n}}}$,

where the Riemann zeta function denotes and the quadratic part of , that is, the greatest quadratic divisor. The number increased from Andrew Granville and Olivier Ramare for upper bound be determined (1996). In connection with an earlier proof of Erdős' conjecture for , this was thus generally proven. ${\displaystyle \zeta }$ ${\displaystyle s(n)}$ ${\displaystyle n}$ ${\displaystyle 2^{8000}}$ ${\displaystyle n_{0}}$ ${\displaystyle 4<n<2^{774840978}}$

Individual evidence

1. ↑ Eric Weis Stone: Erdős Square Free Conjecture . In: MathWorld (English).
2. ↑ Eric Weisstein: Sárkőzy's Theorem . In: MathWorld (English).