By Abraham A. Fraenkel
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Additional resources for Integers and Theory of Numbers
FRAENKEL 41 with the aid of ruler and compass alone it is possible to divide the circle into 17, 257, and 65,537 equal parts. Should we discover an additional prime of the same form, we would, thereby, acquire knowledge of a new geometrical fact. (Needless to add that, for practical purposes, no one would em ploy these theoretical methods in dividing the circle into 17 equal parts. ) Let us consider another theorem, which at first glance appears additive in character, but which m ay nevertheless be easily proved owing to the possibility of expressing it m ultiplicatively.
The sum of the digits of 681472 is 28. 19 W hat we have here said regarding 9 is true of 3 as well, for 10 is also congruent to 1 modulo 3. This check, however, is not as powerful since, even if our calculations were false, there would still be a prob ability of 1 /3 that the wrong result is congruent to the correct one modulo 3, while in the case of 9 the probability is only 1/9. Another test m ay be made by the use of 11. Here we rely on the congruences: 10 2= —1 (m od. 11); 100 = + 1 (mod. 11); 1000 = 10,000 = + 1 (mod.
In the theory of numbers, their validity is much less certain than in the experi mental sciences. W e m ay accept a conclusion only after it has been proved mathematically. Let us consider a few instances. Euler found, b y testing all the integers up to 2500, that it was apparently possible to express every odd number as the sum o f a prime number, p, and of the double of a square: n = p + 2m2. A t the time, no one doubted that this was always possible. Later, the calculations in ques tion were extended to cover all odd numbers up to 9000.