 By Weber W.

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Additional info for Die Pellsche Gleichung

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He employed a method much like ours, using the division algorithm repeatedly until a remainder of 1 is reached. Bachet then performed a sequence of “back substitutions” in a special way so as to avoid the need of negative numbers (which were not yet in common use). Leonhard Euler may have been the first to give an actual proof that if a and b are relatively prime, then ax C by D c is solvable in integers. What Euler in fact demonstrated is that the quantities c ka, k D 0; 1; : : : ; b 1 give b distinct remainders when divided by b.

44 before reading further. The answer involves the concept of being relatively prime. 45 Theorem. Let a, b, c and n be integers with n > 0. mod n/. 40 begin to address the question: Given integers a, b, and c, when do there exist integers x and y that satisfy the equation ax C by D c? When we seek integer solutions to an equation, the equation is called a Diophantine equation. 46 Question. Suppose a, b, and c are integers and that there is a solution to the linear Diophantine equation ax C by D c; that is, suppose there are integers x and y that satisfy the equation ax C by D c.

Finally, you reached 7 and discovered that in fact 91 is not a prime. You were probably relieved, as you might have secretly feared that you would have to continue the daunting task of trial division 91 times! The following theorem tells us that you need not have been too concerned. 3 Theorem. A natural number n > 1 is prime if and only if for all primes p p Ä n, p does not divide n. 4 Exercise. Use the preceding theorem to verify that 101 is prime. The search for prime numbers has a long and fascinating history that continues to unfold today.