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# Difference of Powers

For integers $$a, b, m, n$$, prove that $gcd({a}^{m} - {b}^{m}, {a}^{n} - {b}^{n}) = {a}^{gcd(m,n)} - {b}^{gcd(m,n)}.$

Solution

Let $$d = gcd(m,n)$$, thus $$d|m$$ and $$d|n$$. We then let $$m=pd$$ and $$n=pd$$.

Hence $gcd({a}^{m} - {b}^{m}, {a}^{n} - {b}^{n}) = gcd({a}^{pd} - {b}^{pd},{a}^{qd} - {b}^{qd}).$

Let $$A ={a}^{d}$$ and $$B = {b}^{d}$$, it follows that

$gcd({a}^{m} - {b}^{m}, {a}^{n} - {b}^{n}) = gcd({A}^{p} - {B}^{p},{A}^{q} - {B}^{q}).$

Since $\frac{{A}^{p} - {B}^{p}}{A-B} = \sum _{ k=0 }^{ p-1 }{ { A }^{ p-k-1 }{ B }^{ k } }$ and $\frac{{A}^{q} - {B}^{q}}{A-B} = \sum _{ k=0 }^{ q-1 }{ { A }^{ q-k-1 }{ B }^{ k } }$ we get

$gcd({A}^{p} - {B}^{p},{A}^{q} - {B}^{q}) = A-B.$

Since $$A-B = {a}^{d} - {b}^{d}$$ and $$d =gcd(m,n)$$, we prove that

$gcd({a}^{m} - {b}^{m}, {a}^{n} - {b}^{n}) = {a}^{gcd(m,n)} - {b}^{gcd(m,n)}.$

Check out my other notes at Proof, Disproof, and Derivation

Note by Steven Zheng
2 years, 8 months ago

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