A necessary and suffcient condition is given for the equation Ax4+ 1 =By2 to have positive integer solution, and an effective method is derived for solving equation a2x4 + 1 = By2 in positive integers x, y for given h...A necessary and suffcient condition is given for the equation Ax4+ 1 =By2 to have positive integer solution, and an effective method is derived for solving equation a2x4 + 1 = By2 in positive integers x, y for given ho and B completely. Also, using a recently result of Ribet, Darmon and Merel, we proved that Erdos’ conjecture on combinatorial number is right.展开更多
A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. Suppo...A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. Suppose ξ(s) = ξ1(a,b) + iξ2(a,b) = 0 but ζ(s) = ζ1(a,b) + iζ2(a,b) ≠ 0 with s = a + ib at first. By comparing the real part and the imaginary part of Zeta function equation individually, a set of equation about a and b is obtained. It is proved that this equation set only has the solutions of trivial zeros. In order to obtain possible non-trivial zeros, the only way is to suppose that ζ1(a,b) = 0 and ζ2(a,b) = 0. However, by using the compassion method of infinite series, it is proved that ζ1(a,b) ≠ 0 and ζ2(a,b) ≠ 0. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.展开更多
基金Supported by the Natural Science Foundation of Heilongjiang Province
文摘A necessary and suffcient condition is given for the equation Ax4+ 1 =By2 to have positive integer solution, and an effective method is derived for solving equation a2x4 + 1 = By2 in positive integers x, y for given ho and B completely. Also, using a recently result of Ribet, Darmon and Merel, we proved that Erdos’ conjecture on combinatorial number is right.
文摘A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. Suppose ξ(s) = ξ1(a,b) + iξ2(a,b) = 0 but ζ(s) = ζ1(a,b) + iζ2(a,b) ≠ 0 with s = a + ib at first. By comparing the real part and the imaginary part of Zeta function equation individually, a set of equation about a and b is obtained. It is proved that this equation set only has the solutions of trivial zeros. In order to obtain possible non-trivial zeros, the only way is to suppose that ζ1(a,b) = 0 and ζ2(a,b) = 0. However, by using the compassion method of infinite series, it is proved that ζ1(a,b) ≠ 0 and ζ2(a,b) ≠ 0. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.
