91Ó°ÊÓ

Verify that the integers 1949 and 1951 are twin primes.

Employing the Sieve of Eratosthenes, obtain all the primes between 100 and 200 .

Sylvester (1896) rephrased the Goldbach conjecture: Every even integer \(2 n\) greater than 4 is the sum of two primes, one larger than \(n / 2\) and the other less than \(3 n / 2\). Verify this version of the conjecture for all even integers between 6 and 76 .

If \(p \geq 5\) is a prime number, show that \(p^{2}+2\) is composite. [Hint: \(p\) takes one of the forms \(6 k+1\) or \(6 k+5 .]\)

A conjecture of Lagrange ( 1775 ) asserts that every odd integer greater than 5 can be written as a sum \(p_{1}+2 p_{2}\), where \(p_{1}, p_{2}\) are both primes. Confirm this for all odd integers through 75 .

Find the prime factorization of the integers 1234,10140, and 36000 .

It has been conjectured that every even integer can be written as the difference of two consecutive primes in infinitely many ways. For example, $$ 6=29-23=137-131=599-593=1019-1013=\cdots $$ Express the integer 10 as the difference of two consecutive primes in 15 ways.

Prove that a positive integer \(a>1\) is a square if and only if in the canonical form of \(a\) all the exponents of the primes are even integers.

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 \(n=p_{1}^{k_{1}} p_{2}^{k_{4}} \cdots p_{s}^{k_{1}}\) 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.]

Numerical evidence makes it plausible that there are infinitely many primes \(p\) such that \(p+50\) is also prime. List 15 of these primes.