MathDB

53

Part of PEN H Problems

Problems(1)

H 53

Source:

5/25/2007
Suppose that a,ba, b, and pp are integers such that b≡1  (mod4)b \equiv 1 \; \pmod{4}, p≡3  (mod4)p \equiv 3 \; \pmod{4}, pp is prime, and if qq is any prime divisor of aa such that q≡3  (mod4)q \equiv 3 \; \pmod{4}, then qp∣a2q^{p}\vert a^{2} and pp does not divide q−1q-1 (if q=pq=p, then also q∣bq \vert b). Show that the equation x2+4a2=yp−bpx^{2}+4a^{2}= y^{p}-b^{p} has no solutions in integers.
modular arithmeticDiophantine Equations