Some Algebraic Identities for use in number theory — RMO

Many problems about divisibility can be solved using algebraic identities such as:

  1. {}a-b|a^{n}-b^{n} for all n \in N and a+b|a^{n}+b^{n} for all odd n \in N.
  2. Note that 1+a+\ldots + a^{n}=\frac{a^{n+1}-1}{a-1} provided that n \neq 1.
  3. (Sophie Germain’s identity)

a^{4}+4b^{4}=a^{4}+4a^{2}b^{2}+4b^{4}-4a^{2}b^{2}=(a^{2}+2b^{2})^{2}-(2ab)^{2} = (a^{2}+2ab+2b^{2})(a^{2}-2ab+2b^{2})

Here is a problem which can be solved using algebraic identities:

Example: 

Show that n^{5}+n^{4}+1 is not a prime for n>1.

Solution:

Note that

n^{5}+n^{4}+1=n^{5}+n^{4}+n^{3}+n^{2}+n+1-(n^{3}+n^{2}+n)

which in turn equals

\frac{n^{6}-1}{n-1} - n \frac{n^{3}-1}{n-1}=(n^{3}+1)\frac{n^{3}-1}{n-1}-n\frac{n^{3}-1}{n-1}, that is,

=\frac{(n^{3}-1)(n^{3}-n+1)}{n-1} = (n^{2}+n+1)(n^{3}-n+1)

Since n>1, each n^{2}+n+1, n^{3}-n+1 is greater than 1. Hence, n^{5}+n^{4}+1 is not prime for n>1.

More later,

Nalin Pithwa

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