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Question

If pbe a natural number, then prove that pn+1+(p+1)2n1 is divisible by p2 +p + 1 for every positive integer n.

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Solution

Let f (n) = pp+1+(p+1)2n1
We have f (1) = p2 + p + 1 so that f (1) is divisible by p2 + p + 1.
Now assume that f (m) is divisible by p2 + p + p + 1 i.e.,
we assume that
pm+1+(p+1)2m1=k(p2+p+1)
Now f (m + 1) = pm+2+p(p+1)2m+21
=pm+2++[k(p2+p+1)pm+1](p+1)2
=pm+2(p+1)2pm+1+k(p+1)2(p2+p+1)
=pm+1(pp22p1)+k(p+1)2(p2+p+,1)
=(p2+p+1)[k(p+1)2pm+1]
Hence f (m + 1) is divisible by p2 + p + 1.
By induction, f (x) is divisible by p2 + p + 1 for all n ϵ N.

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