query: A property of divisibility by an odd prime $p$
positive_passages.docid: 148458
positive_passages.text: $x^p-x \equiv x(x-1)(x-2)\cdots (x-(p-1))\,\pmod{\!p}$
negative_passages.docid: 49711
negative_passages.text: Prove that $6p$ is always a divisor of $ab^{p} - ba^{p}$.
negative_passages.docid: 68328
negative_passages.text: Infinitely many $n$ such that $2^n-n$ is divisible by a prime
negative_passages.docid: 92416
negative_passages.text: About a sum of Legendre symbols associated to an odd prime
negative_passages.docid: 65764
negative_passages.text: Partitioning $\{1,2,\ldots,k\}$ into $p$ subsets with equal sums, where $p$ is prime
negative_passages.docid: 50781
negative_passages.text: $(n+1)$th prime $p_{n+1}$ less than or equal to $p_1p_2\dots p_n+1$
negative_passages.docid: 46431
negative_passages.text: Proof that $\sum\limits_i \frac{(p-1)!}i$ is divisible by $p$
negative_passages.docid: 99205
negative_passages.text: Group of order $p^k q$ where $p$ and $q$ are distinct primes - how to prove this property?
negative_passages.docid: 63893
negative_passages.text: Prime dividing the binomial coefficients
negative_passages.docid: 75516
negative_passages.text: Relatively Prime Numbers and Number of Divisors of n
negative_passages.docid: 144814
negative_passages.text: If $p$ and $q$ are primes, which binomial coefficients $\binom{pq}{n}$ are divisible by $pq$?
negative_passages.docid: 5841
negative_passages.text: Proving prime $p$ divides $\binom{p}{k}$ for $k\in\{1,\ldots,p-1\}$
negative_passages.docid: 119630
negative_passages.text: For an odd prime $p$, $\;p =1 \pmod 4,\; \;x^2+1 = 0 \pmod p$
negative_passages.docid: 5978
negative_passages.text: Prove that there exists only one prime number of the form $p^2−1$ where $p≥2$ is an integer
negative_passages.docid: 120554
negative_passages.text: highest power of prime $p$ dividing $\binom{m+n}{n}$
negative_passages.docid: 124890
negative_passages.text: Divisibility within a block of natural numbers
negative_passages.docid: 156607
negative_passages.text: Show that $L_p =\frac{1^p+2^p+3^p+...+(p-1)^p}{p}$ is an integer and generalize the integer for all odd integers p