Download (2s1) Designs withs intersection numbers by Ionin Y. J., Shrikhande M. S. PDF

By Ionin Y. J., Shrikhande M. S.

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26 A. Diophantine equations and integral forms by formula (12) we have Corollary 2. Under a birational G transformation of the form (10), which gives the equivalence of (1) and (9), every class of solutions of equation (9) maps to a class of solutions of equation (1). Now suppose that in equation (1) we have (ϑn , ϑ1 · · · · · ϑn−1 ) = 1 (n > 1). Therefore mn = (ϑn , [ϑ1 , . . , ϑn−1 ]) = 1, and by Theorem 1 equation (1) is equivalent by a birational G transformation to the equation n−1 Ak Xkmk + An Xn = 0, (14) k=1 all of whose rational solutions are given by the formulas: Xk = tk (1 k n − 1), Xn = − mk n−1 k=1 Ak tk An , where tk (1 k n − 1) are rational parameters.

M) be the conjugates of ω, where m is the degree of J . Since J is contained in Q(θ ), we have ω = g(θ ), where g is a polynomial with rational coefficients. Thus G(x) has a zero in common with the polynomial m g(x) − ω(j ) , (13) j =1 which has rational coefficients, and since G(x) is irreducible, it must divide this polynomial. The factors of (13) are relatively prime in pairs, since their differences are non-zero constants. Hence the polynomials H (j ) (x) = G(x), g(x) − ω(j ) are relatively prime in pairs, and since each of them divides G(x), their product must (1 ) In dealing with this case we do not need to exclude the possibility that (ej , n) = 1.

Vn ) for some rational v1 , . . , vn . Thus when K is cyclic, we have three apparently different conditions on f (x) which are in reality equivalent. Corollary to Theorem 2. Let f (x) be a polynomial with integral coefficients, and suppose that every arithmetical progression contains an integer x such that f (x) is a sum of two squares. Then f (x) = u21 (x) + u22 (x) identically, where u1 (x) and u2 (x) are polynomials with integral coefficients. In the particular case of Theorem 2, namely the case K = Q(i), which is needed for this Corollary, our method of proof has much in common with that used by Lubelski [7] in his investigation of the primes p for which f (x) ≡ 0 (mod p) is soluble.

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