关于Hall问题

On a Problem of Hall

本文我们进一步得到方程q^m=p^n+2,p,q是素数,m>1,n>1(*)无解的一些结果:(1)设q=p+4+6a或p=q+2+6a,则除开3~3=5~2+2以外方程(*)无解.设ε=u+v(p^(1/P)是pll方程x^2-py^2=1的基本解,我们有(2)设q=p+6a或p=q+6a,则当u≠1/2(q^m+p^n)或v≠q^(m/2)p^(n-1/2)时方程无解.(3)设q=p+2+6a,p>3,a≠0(mod4)或p=q+4+6a,q>3,a≠3(mod4),则当u≠1/2(q^m+p^n)或v≠q(m/2)p(n-1/2)时方程(*)无解.

In this paper, let ε=u +vp be fundamental solution of the Pell equation x2-py2= 1, we give elementary proof of the following theorems: (l) If q =p + 4 + 6a or p = q + 2 + 6a, then the equation qm =pn+ 2, p, q are primes, m > 1,n > 1(*) has no solution except 33= 52+2. u≠1/2(qm+pn) or v≠ q2/m then the equation (2) If q =p +6a or p = q + 6a, u /, (q' +p') dr v /q2 p2,then the equation (*) has no solution (3) If q =p +2+6a, p>3, a 0 (mod4) or p =q +4+6a, q >3, a 3 (mod4), u 1/2(qm+ pn) or v q 2 + P2,then the equation (*) has no solution. (4) If q = pt2 1,q = pt2+ 4 or p = qt2 1,p = qt2-4, then the equation (*) has no so- lution.