Contributions to the Founding of the Theory of Transfinite by George Cantor, Philip Edward Bertrand Jourdain

By George Cantor, Philip Edward Bertrand Jourdain

Covers addition, multiplication and exponentiation of cardinal numbers, smallest transfinite cardinal numbers, ordinal different types of uncomplicated ordered aggregates and operations on ordinal varieties. Develops conception of well-ordered aggregates; investigates ordinal numbers of well-ordered aggregates and extra.

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2 Linear congruences We now return to the question of division of congruence classes, postponed from earlier in this chapter. In order to assign a meaning to a quotient [b]/[a] of two congruence classes [aJ, [b] E :In, we need to consider the solutions of the linear congruence ax == b mod (n). Note that if x is a solution, and if x' == x, then ax' == ax == b and so x' is also a solution; thus the solutions (if they exist) form a union of congruence classes. Now ax == b mod (n) if and only if ax - b 47 3.

If m is not a power of 2, then m has the form 2nq for some odd q > 1. Now the polynomial f(t) = t q + 1 has a root t = -1, so it is divisible by t + 1; this is a proper factor since q > 1, so putting t = x2n we see that the polynomial g(x) = f(X 2R ) = xm + 1 has a proper factor x2n + 1. Taking x = 2 we see that 22R + 1 is a proper factor of the integer g(2) = 2m + 1, which cannot 0 therefore be prime. Numbers of the form Fn = 22n + 1 are called Fermat numbers, and those which are prime are called Fermat primes.

0 For example, we can see that 97 is prime by checking that it is divisible by none of the primes p ~ V97, namely 2, 3, 5 and 7. This method requires us to test whether an integer n is divisible by various primes p. For certain small primes p there are simple ways of doing this, based on properties of the decimal number system. In decimal notation we write a positive integer n in the form akak-l ... alaO, meaning that where ao, ... , ak are integers with 0 ::; ai ::; 9 for all i, and ak =f:. o.

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