proof-or-given-a-counter-example-if-p-is-prime-and-a-N-then-p-a-p-p-a-p-

Question Number 469 by 123456 last updated on 25/Jan/15

p r o o f o r g i v e n a c o u n t e r - e x a m p l e

i f p i s p r i m e a n d a \in N t h e n

p ∣ {(a + p)}^{p} - a^{p}

Answered by prakash jain last updated on 10/Jan/15

{(a + p)}^{p} = \underset{i = 0}{\sum^{p}}^{p} C_{i} a^{i} p^{p - i}

{(a + p)}^{p} - a^{p} = \underset{i = 0}{\sum^{p - 1}}^{p} C_{i} a^{i} p^{p - i}

All terms on the expansion in RHS

include p as a factor .

Hence p ∣ {(a + p)}^{p} - a^{p}