query: Show that no integer of the form $a^3 +1$ is a prime for $a>1$
positive_passages.docid: 69136
positive_passages.text: factorization of a^n+1?
negative_passages.docid: 152299
negative_passages.text: Proving $a^{n-1} \equiv 1 \mod n $ when n is not prime.
negative_passages.docid: 75206
negative_passages.text: If $a$, $a+2$ and $a+4$ are prime numbers then, how can one prove that there is only one solution for $a$?
negative_passages.docid: 44857
negative_passages.text: Why the $(n+1)$th prime number cannot exceed the smallest prime divisor of $p_1\dotsc p_n + 1$
negative_passages.docid: 130342
negative_passages.text: Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
negative_passages.docid: 78343
negative_passages.text: $4x^2+1$ factors only into $4y+1$ primes
negative_passages.docid: 51382
negative_passages.text: Proving that either $2^n-1 $ or $ 2^n+1$ is not prime
negative_passages.docid: 77070
negative_passages.text: Prove Euler's Theorem when the integers are not relatively prime
negative_passages.docid: 123296
negative_passages.text: Determine whether a number is prime
negative_passages.docid: 39915
negative_passages.text: Let $n$ be an integer such that if $d | n$ then $d + 1 | n + 1.$ Show that $n$ is a prime number
negative_passages.docid: 57050
negative_passages.text: $\frac{a^{3}+1}{b+1}+\frac{b^{3}+1}{a+1}$ an integer $\Rightarrow \frac{a^{3}+1}{b+1}$ and $\frac{b^{3}+1}{a+1}$ are integers.
negative_passages.docid: 47162
negative_passages.text: Proving there are infinitely many primes of the form $a2^k+1.$
negative_passages.docid: 84379
negative_passages.text: Prove that there is no integer a for which $a^2 - 3a - 19$ is divisible by 289
negative_passages.docid: 151701
negative_passages.text: Does there exist an $a_0$ such that the sequence $a_{n+1} = 2a_n + 1$ is prime for all $n \ge 0$?
negative_passages.docid: 79544
negative_passages.text: Prove $x^3 + 3 =4y(y+1)$ has no integer solutions
negative_passages.docid: 140006
negative_passages.text: Are there infinitely many primes and non primes of the form $10^n+1$?
negative_passages.docid: 111935
negative_passages.text: Irreducible but not prime