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186 = -39 + 143t - 30t, For instance, putting t = -1 we obtain the solution consisting of the smallest numbers in absolute value, namely x = 43, Y = -9. Sometimes we are only interested in solutions within a certain range. 54 For instance, suppose that in the above example we want to find all positive solutions. Thus we need to find all those values of t for which -39 - 30t> 0 186 + 143t > O. and The first of these inequalities implies that t second implies that t ~ -1. ~ -2 whilst the Thus in this case we see that there are in fact no positive solutions.

0 THE EUCLIDEAN ALGORITHM The notion of divisibility of one number by another is fundamental to practically all aspects of Number Theory. Given any two numbers one can add them or multiply them and obtain a new (natural) number. If you allow for negative numbers (and zero), by considering the integers rather than just the positive integers, you can subtract as well. But division cannot, in general, be performed, which is to say the result of dividing one number* or integer by another is not necessarily another number* or integer.

Enumerates the primes in order, are Nor is it known if This is not known. (Presumably the answer infinitely many of them are composite. ) 6. DIOPHANTINE EQUATIONS In honour of the Ancient Greek mathematician Diophantus, we use the name Diophantine Equation to refer to an equation with integer coefficients for which a solution is sought in the integers. The simplest non-trivial form of Diophantine equation is the linear equation in two variables: ax + by c, where a,b,c are integers and integer solutions for x,y are sought.

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Addition theorems; the addition theorems of group theory and number theory by Henry B. Mann.

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