. Carnegie Institution of Washington publication. CHAP. II] NUMBER OF SOLUTIONS OF ax+by = 69 V. Bernard!135 would find the positive integral solutions of ax + by = k by employing the remainders r[, r[' and quotients q[, q\' obtained on dividing k — b by a and k — a by b. Thus axi + byi = kit ki = k — a — b — r( — r". Similarly, ax2 + by2 = fca, • • •, axm + bym = km = km-i — a — b — r'm — r^, where r'm, r'^ are the remainders and qm, q^. the quotients obtained on dividing km-i — b by a and km-i — a by b. In this way we find a value uoim such that a zero remainder results from that one of
. Carnegie Institution of Washington publication. CHAP. II] NUMBER OF SOLUTIONS OF ax+by = 69 V. Bernard!135 would find the positive integral solutions of ax + by = k by employing the remainders r[, r[' and quotients q[, q\' obtained on dividing k — b by a and k — a by b. Thus axi + byi = kit ki = k — a — b — r( — r". Similarly, ax2 + by2 = fca, • • •, axm + bym = km = km-i — a — b — r'm — r^, where r'm, r'^ are the remainders and qm, q^. the quotients obtained on dividing km-i — b by a and km-i — a by b. In this way we find a value uoim such that a zero remainder results from that one of the two divisions in which the divisor is the smaller of a, b, or such that the remainder from the other division is zero or is divisible by the smaller coefficient. Then ku is divisible by the larger or the smaller of a, b in the respective cases. The positive integral solutions with ku divisible by a are xu = ku/a — nb, yu = na (n = 0, 1, • • • Then all positive integral solutions of the given equation are = ( xu + q[ - Cf. L. Crocchi136 noted that Hermite's119 formulas do not give merely the integral solutions. Thus, if n < a, n < b, they give x = ± b£, y = =fc ar], % + 77 = ± n/(ab), which lead to fractional solutions of ax + by = n. Crocchi therefore transformed Hermite's formulas so that the resulting formulas give merely positive integral solutions. Set n' = n — r — s, Then 1= ? .-= ^.. - =^ a" LaJ a La \ La J a' La J LaJ where [s/a]+ is the quotient by excess of s by a. Similarly, r^i = [?]_[!] _[£] . L a J La J LaJ+ La J+ Taking alternate signs and adding, we get, for m even, . LaJ 1L«J+"LTJ+ a+. + 135 Atti society italiana per il progresso delle scienze, 2, 1908, 317-8. »«II Boll, di Matematica Gior. , 7, 1908, Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these ill
Size: 1429px × 1749px
Photo credit: © Book Worm / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookauthorcarnegie, bookcentury1900, bookdecade1900, bookyear1902
