By D. J. Struik
These chosen mathematical writings conceal the years whilst the principles have been laid for the speculation of numbers, analytic geometry, and the calculus.
Originally released in 1986.
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Extra resources for A Source Book in Mathematics, 1200-1800
And in general, if it is known that A < (p — 1)/», then one proves in the same way that we cannot have λ > (ρ — 1 )/(n + 1), therefore we must have λ = (ρ — 1)/(w + 1) or λ < (ρ — 1)(n + 1). 47. Corollary 3. Wherefrom it appears that the number of all numbers that cannot be residues is either = 0, or = λ, or = 2A or any multiple of A; for if there are more than ηλ of such numbers, then, if any at all, A new ones are added to them, so as to make their number = (η + 1)A; and if this does not yet comprise all the nonresidues, then at once A new ones are added.
I. We may consider the numbers χ and y as prime to each other; for if they had a common divisor, the cubes would also be divisible by the cube of that divisor. For example, let χ = 2α, and y = 26, we shall then have x 3 + y 3 = 8a3 + 863; now if this formula be a cube, a3 + 63 is a cube also. II. Since, therefore, χ and y have no common factor, these two numbers are either both odd, or the one is even and the other odd. In the first case, ζ would be even, and in the other that number would be odd.
On the enormous literature in this field see P. Bachman, Das Fermatproblem (De Gruyter, Berlin-Leipzig, 1919); L. J. Mordell, Three lectures on Fermat's last theorem (Cambridge University Press, Cambridge, England, 1921); It. Nogues, Theoreme de Fermat. Son histoire (Vuibert, Paris, 1932); H. S. Vandiver, "Fermat's last theorem," American Mathematical Monthly S3 (1946), 555-578. 9) how Euler proved Fermat's theorem for η = 3 and η = 4. Fermat communicated many of his results to the mathematician Bernard Frenicle de Bessy (1605-1675).