query: Show that 13 is the largest prime which divides two consecutive terms of $n^2 + 3$.
positive_passages.docid: 157914
positive_passages.text: $13$ is the largest prime that can divide two successive integers of the form $n^2+3$
negative_passages.docid: 120554
negative_passages.text: highest power of prime $p$ dividing $\binom{m+n}{n}$
negative_passages.docid: 50663
negative_passages.text: prove or disprove: For a prime number p, there is a positive integer n such that $p+1 $= $n^2$ if and only if $ p = 3.$
negative_passages.docid: 103784
negative_passages.text: Show that $n^2+11n+2$ is not divisible by $113^2$ ( n is integer)
negative_passages.docid: 6700
negative_passages.text: An Olympiad problem in prime numbers :can we proof that "n" is prime in this case "
negative_passages.docid: 88656
negative_passages.text: For $p$ prime, show that $\Phi_{p^n}(x) = 1 + x^{p^{n-1}} + x^{2p^{n-2}} + \dots +x^{(p-1)^{p^n-1}}$
negative_passages.docid: 91914
negative_passages.text: Prove that for the Fibonacci sequence $(F_n)$, $F_n$ divides $F_{2n}$.
negative_passages.docid: 38904
negative_passages.text: Prove that $(a + b + c)^{13}$ is divisible by $abc$ if $b|a^3$, $c|b^3$ and $a|c^3$.
negative_passages.docid: 34588
negative_passages.text: Number Theory: $p!+p=n^2$, $p$ = prime
negative_passages.docid: 37330
negative_passages.text: Prove that any two consecutive terms of the Fibonacci sequence are relatively prime
negative_passages.docid: 24053
negative_passages.text: How can I prove that $n^7 - n$ is divisible by $42$ for any integer $n$?
negative_passages.docid: 71650
negative_passages.text: Proving by strong induction for a sequence of integers, $2^n$ divides term $n$
negative_passages.docid: 15348
negative_passages.text: prime divisor of $3n+2$ proof
negative_passages.docid: 40896
negative_passages.text: What is the largest positive $n$ for which $n^3+100$ is divisible by $n+10$
negative_passages.docid: 26503
negative_passages.text: Suppose that $p$ ≥ $q$ ≥ $5$ are both prime numbers. Prove that 24 divides ($p^2 − q^2$)
negative_passages.docid: 50534
negative_passages.text: Prove that if $n^2$ is divided by 3, then also $n$ can also be divided by 3.
negative_passages.docid: 97666
negative_passages.text: Prove $133\mid \left(11^{n+2}+12^{2n+1}\right)$