91Ó°ÊÓ

Determine whether the integer 701 is prime by testing all primes \(p \leq \sqrt{701}\) as possible divisors. Do the same for the integer 1009 .

In 1752 , Goldbach submitted the following conjecture to Euler: Every odd integer can be written in the form \(p+2 a^{2}\), where \(p\) is either a prime or 1 and \(a \geq 0\). Show that the integer 5777 refutes this conjecture.

Show that any composite three-digit number must have a prime factor less than or equal to 31 .

Find all prime numbers that divide \(50 !\).

An integer is said to be square-free if it is not divisible by the square of any integer greater than 1. Prove the following: (a) An integer \(n>1\) is square-free if and only if \(n\) can be factored into a product of distinct primes. (b) Every integer \(n>1\) is the product of a square-free integer and a perfect square. [Hint: If \(n=p_{1}^{k_{1}} p_{2}^{k_{2}} \cdots p_{s}^{k_{1}}\) is the canonical factorization of \(n\), then write \(k_{i}=2 q_{i}+r_{i}\) where \(r_{i}=0\) or 1 according as \(k_{i}\) is even or odd.]

Show that 13 is the largest prime that can divide two successive integers of the form \(n^{2}+3\)

(a) The arithmetic mean of the twin primes 5 and 7 is the triangular number 6 . Are there any other twin primes with a triangular mean? (b) The arithmetic mean of the twin primes 3 and 5 is the perfect square \(4 .\) Are there any other twin primes with a square mean?

Prove that for every \(n \geq 2\) there exists a prime \(p\) with \(p \leq n<2 p\). [Hint: In the case where \(n=2 k+1\), then by the Bertrand conjecture there exists a prime \(p\) such that \(k