Prime Numbers

1. Prove that for every integer k the numbers 2k+l and 9k+4 are relatively

prime, and for numbers 2k-l and 9k+4 find their greatest common

divisor as a function of k.

2. Prove that there exists an increasing infinite sequence of triangular

numbers (i.e. numbers of the form t_n = 0.5*n(n+ 1), n = 1, 2, ... ) such that

every two of them are relatively prime.

3. Prove that there exists an increasing infinite sequence of tetrahedral

numbers (i.e. numbers of the form T_n = (1/6)*n(n+ 1)(n+2),( n = 1,2, ... ), such

that every two of them are relatively prime.

4. Prove that if a and b are different integers, then there exist infinitely

many positive integers n such that a+n and b+n are relatively prime.

5 . Prove that if a, b, c are three different integers, then there exist infinitely

many positive integers n such that a+n, b+n, c+n are pairwise relatively

prime.

6. Give an example of four different positive integers a, b, c, d such

that there exists no positive integer n for which a+n, b+n, c+n, and d+n

are pairwise relatively prime.

7. Prove that every integer> 6 can be represented as a sum of two

integers > 1 which are relatively prime.

8*. Prove that every integer > 17 can be represented as a sum of three

integers > 1 which are pairwise relatively prime, and show that 17 does not

have this property.

9*. Prove that for every positive integer m every even number 2k can be

represented as a difference of two positive integers relatively prime to m.

10*. Prove that Fibonacci's sequence (defined by conditions U₁ = U₂

= 1, U_n+2 = U_n+U_n+1, n = 1, 2, ... ) contains an infinite increasing sequence

such that every two terms of this sequence are relatively prime.