By F. Ruggiero

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**Example text**

Similarly. we define three numbers to be relatively prime when their greatest common divisor is 1. They are relatively prime if and only if there exists no prime which is a common divisor of all three. 18 1. 14 7. AN ALTERNATE FOUNDATION OF THE THEORY OF THE GREATEST COMMON DIVISOR. DEFINITION A non-empty set of complex numbers is called a modu1e* if it is closed under addition and subtraction; that is. the sum and difference of two elements of the set is an element of the set. For example. the set {O} is a module.

15; the number f2(n) = n(n+1) + 41 is prime for n = 0, I, 2, ·.. , 39; the number f3(n) = 2n2 + 29 is prime for n = 0, I, 2, ·.. , 28. * For historical reasons, this is called Bertrand's postulate. t Dirichlet's theorem on primes in an arithmetic progression. 4). To conclude this section we present three questions to which, at this time, no answer is known. I. Two primes p,q with difference p-q example, {3,5}, {11,13}, {17,19}, The unanswered question is: = 2 are called twin primes. , {l49,151}, ••• are twin primes • Is the number of twin primes finite or i nfi nHe?

For n = 18 and d = 2 the number of those integers among 1. 2••••• 18 which have greatest common divisor 2 with 18 is ~(l~J = ~(9) = 9(1- t) = 6; and they are 2. 4. 8. 10. 14. 16. THEOREM 3 For every natural number n we have n = E din ~(d) • Proof Let d1 = 1. d2• d3 ••••• dk = n be the divisors of n in ascending order. The set of integers {I. 2••••• n} may be classified into subsets in the following way: 1st subset: integers which have GCD 1 with n 2nd subset: integers which have GCD d2 with n jth subset: integers which have GCD dj with n kth subset: integers which have GCD dk=n with n.