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Luka
Luka
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$p^3 $n^3 | \sum_{i=1}^{pn-1}\binom{pn}{i}^2$ => $p$n | \sum_{i=1}^{pn-1}\binom{pn}{i}$?

For $p\in \mathbf{N}$$n\in \mathbf{N}$ is $$p^3 \text{ divides } \sum_{i=1}^{p-1}\binom{p}{i}^2$$$$n^3 \text{ divides } \sum_{i=1}^{n-1}\binom{n}{i}^2=\binom{n}{1}^2+\cdots +\binom{n}{n-1}^2$$ impling $$p \text{ divides } \sum_{i=1}^{p-1}\binom{p}{i}$$$$n \text{ divides } \sum_{i=1}^{n-1}\binom{n}{i}=\binom{n}{1}+\cdots +\binom{n}{n-1}$$?

Notice that the second condition is equvalentequivalent to say: "$p$$n$ is a prime or $p$$n$ is a Poulet number". Or

is $$p^4 \text{ divides } \sum_{i=1}^{p-1}\binom{p}{i}^2$$$$n^4 \text{ divides } \sum_{i=1}^{n-1}\binom{n}{i}^2=\binom{n}{1}^2+\cdots +\binom{n}{n-1}^2$$ impling $$p \text{ divides } \sum_{i=1}^{p-1}\binom{p}{i}$$$$n \text{ divides } \sum_{i=1}^{n-1}\binom{n}{i}=\binom{n}{1}+\cdots +\binom{n}{n-1}$$? Notice that the first condition here is equvalent to say: "$p$$n$ is a Wolstenholme number"

$p^3 | \sum_{i=1}^{p-1}\binom{p}{i}^2$ => $p | \sum_{i=1}^{p-1}\binom{p}{i}$?

For $p\in \mathbf{N}$ is $$p^3 \text{ divides } \sum_{i=1}^{p-1}\binom{p}{i}^2$$ impling $$p \text{ divides } \sum_{i=1}^{p-1}\binom{p}{i}$$?

Notice that the second condition is equvalent to say: "$p$ is a prime or $p$ is a Poulet number". Or

is $$p^4 \text{ divides } \sum_{i=1}^{p-1}\binom{p}{i}^2$$ impling $$p \text{ divides } \sum_{i=1}^{p-1}\binom{p}{i}$$? Notice that the first condition here is equvalent to say: "$p$ is a Wolstenholme number"

$n^3 | \sum_{i=1}^{n-1}\binom{n}{i}^2$ => $n | \sum_{i=1}^{n-1}\binom{n}{i}$?

For $n\in \mathbf{N}$ is $$n^3 \text{ divides } \sum_{i=1}^{n-1}\binom{n}{i}^2=\binom{n}{1}^2+\cdots +\binom{n}{n-1}^2$$ impling $$n \text{ divides } \sum_{i=1}^{n-1}\binom{n}{i}=\binom{n}{1}+\cdots +\binom{n}{n-1}$$?

Notice that the second condition is equivalent to say: "$n$ is a prime or $n$ is a Poulet number". Or

is $$n^4 \text{ divides } \sum_{i=1}^{n-1}\binom{n}{i}^2=\binom{n}{1}^2+\cdots +\binom{n}{n-1}^2$$ impling $$n \text{ divides } \sum_{i=1}^{n-1}\binom{n}{i}=\binom{n}{1}+\cdots +\binom{n}{n-1}$$? Notice that the first condition here is equvalent to say: "$n$ is a Wolstenholme number"

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Luka
Luka
  • 121
  • 6

$p^3 | \sum_{i=1}^{p-1}\binom{p}{i}^2$ => $p | \sum_{i=1}^{p-1}\binom{p}{i}$?

For $p\in \mathbf{N}$ is $$p^3 \text{ divides } \sum_{i=1}^{p-1}\binom{p}{i}^2$$ impling $$p \text{ divides } \sum_{i=1}^{p-1}\binom{p}{i}$$?

Notice that the second condition is equvalent to say: "$p$ is a prime or $p$ is a Poulet number". Or

is $$p^4 \text{ divides } \sum_{i=1}^{p-1}\binom{p}{i}^2$$ impling $$p \text{ divides } \sum_{i=1}^{p-1}\binom{p}{i}$$? Notice that the first condition here is equvalent to say: "$p$ is a Wolstenholme number"