Projections of the globe appropriate for laboratory methods of studying the general circulation of the atmosphere

Cleveland Abbe
1907 Bulletin of the American Mathematical Society  
great as R. The method may be applied to tcR, and we have the following theorem : THEOREM : If no denominator of odd rank, after the first, in the complete quotients obtained by expanding i//cR in a continued fraction, turns out to be a perfect square, the expansion being carried out until the numerator of the last convergent is greater than KR, then the factors of KR differ by more than 2 VKR. As an example of the application of this theorem to the discovery of prime factors, take Jevons's *
more » ... mber 8616460799 == R. No perfect square appears in the denominators of the complete quotients obtained in expanding V R, whence one infers that the factors differ by more than 608 = 2VR. Similar failure attends the expansion of j/2R, j/6R, and i/30 i?. On expanding y210R, however, the third denominator is found to be the square 11881 = 109 2 . The numerator of the second convergent is 2690321 ; we know then that the numbers 2690321 ± 109 contain the desired factors of the given number. The factors are 89681 and 96079. The most advantageous value of K to take is the product of the smallest distinct primes. Thus if /c = 30 and the factors of JR are p and q, then the factors will be discovered if p -30q, 2p -15g, 3p -lOq, or 5p -6q are less than 2 A/30JB.
doi:10.1090/s0002-9904-1907-01512-4 fatcat:vqolrmxhgjgs7pt22pmmsyemxm