By D. J. Struik
These chosen mathematical writings disguise the years whilst the principles have been laid for the idea of numbers, analytic geometry, and the calculus.
Originally released in 1986.
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Additional resources for A Source Book in Mathematics, 1200-1800
EULER. POWER RESIDUES 8 | 33 40. Corollary 3. We have already seen in paragraph 15 that the exponent of this lowest power is necessarily smaller than p. Thus either or in which case, when we know now that either Therefore there is no number between and which could ever have the value 41. Theorem 11. Let p be a prime number and the lowest power of a which, divided by p, gives the residue 1; let _ then the exponent cannot be greater than (p — l)/3; thus either Proof. Since is the lowest power of a which, divided by p, has the residue 1, there are at most different numbers of residues in the series wheneach term is divided by p.
P + a):(B + A) = 3:2. These consequences can all be found in the translation of Pascal's paper in Smith, Source book, pp. 74-75. 26 I ARITHMETIC I For example, let it be proposed to find the number of the cell ξ of the fifth perpendicular rank and of the third parallel rank. Having taken all the numbers that precede the index of the perpendicular rank 5, that is, 1, 2, 3, 4, take as many natural numbers beginning with the index of the parallel rank 3, that is, 3, 4, 5, 6. Now multiply the first numbers into each other, and let the product be 24.
The proof runs as follows. First it is proved by means of the binomial expansion of (1 + l ) " " 1 that p is divisible by p if p is an odd prime. Then by a similar expansion of (1 + a) it is shown that (1 + aY - (1 + a) - (a" - a) = 0 (mod p) if a is not a multiple of p. Since 2" - 2 = 0, the proof follows by complete induction. 36 I I ARITHMETIC 9 EULER. 6, 7). Fermat's theorem that xn + yn = Zn cannot be solved for positive integers x, y, ζ, η, η > 2, attracted him and he gave proofs for the cases η = 3 and η = 4.
A Source Book in Mathematics, 1200-1800 by D. J. Struik