关于丢番图方程2py^2=2x^3+3x^2+x的解

On the Solution of Diophantine Equation 2py^2= 2x^3+ 3x^2+ x

本文运用初等数论简单同余法、分解因子法及反证法等,得到丢番图方程2py2=2x3+3x2+x,(p为素数)无正整数解的情况.(1)当p≡1(mod 8),p≡5(mod 8),p≡7(mod 8)时,则方程无正整数解;(2)当p≡3(mod 8)时,Un+Vnp(1/2)=(x0+y0p(1/2))n.其中x0,y0是Pell方程x2-py2=1的基本解,当n≡0(mod 2)时,则方程无整数解;当n≡1(mod 2)时,若2|x0,则方程无整数解.特别是p≡3(mod 8)且p<100时,2|x0,则方程无整数解.

The conditions that the Diophantine equation 2py^2= 2x^3＋ 3x^2＋X （p is a prime p 〉 3） has no positive integral solutions by the method of simple congruence, decomposition and reduction to absurdity in elementary number theory are shown. （1） In the ease of p≡1 （mod8）,p≡5 （mod8）,p≡7 （mod8）, the equation has no positive integral solutions; （2） In the ease of p≡3（mod8） and Un＋ Vn√P- = （x0＋y0√p）^n, where x0, y0 is a fundamental solution of Pell＇s equation x^2- py^2 = 1, the equation has no positive integral solutions for n≡0（mod2）; In the ease of n≡1（mod 2） and when 21 x0 is obtained, the equation has no positive integral solutions. Especially, if p 〈 100 and when 2Ix0 is obtained, the equation has no positive integral solutions.