Amicable Pair with Smallest Common Prime Factor 5

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Theorem

The smallest known amicable pair whose smallest common prime factor is greater than $3$ is the one whose elements are:

$m_1 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 101 \times 8643 \times 1 \, 947 \, 938 \, 229$

and:

$m_2 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 365 \, 147 \times 47 \, 303 \, 071 \, 129$


This is the smallest counterexample to the observation that:

most amicable pair consist of even integers
most of the rest, whose elements are odd, have both elements divisible by $3$.


Proof

It is to be demonstrated that these numbers are amicable.

From Sigma Function of Integer:

$\displaystyle \sigma \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$

where:

$\displaystyle n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i} = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$

is the prime decomposition of $n$.


When $k_i = 1$ the individual factor becomes $\dfrac {p_i^2 - 1} {p_i - 1} = \dfrac {\left({p_i + 1}\right) \left({p_i - 1}\right)} {p_i - 1} = p_i + 1$.


First we make sure we have all the prime factors:

\(\displaystyle 8643\) \(=\) \(\displaystyle 3 \times 43 \times 67\) $\quad$ $\quad$
\(\displaystyle 1 \, 947 \, 938 \, 229\) \(=\) \(\displaystyle 3^2 \times 739 \times 292 \, 879\) $\quad$ $\quad$
\(\displaystyle 47 \, 303 \, 071 \, 129\) \(=\) \(\displaystyle 67 \times 127 \times 5 \, 559 \, 181\) $\quad$ $\quad$


All other factors given are indeed prime.


We establish the contributions to the $\sigma$ values of $m_1$ and $m_2$ by taking the prime factors in turn, and extracting the prime factors of each result.

First, the elements common to both:

\(\displaystyle \sigma \left({5}\right)\) \(=\) \(\displaystyle 6\) $\quad$ Sigma Function of Prime Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 3\) $\quad$ $\quad$


\(\displaystyle \sigma \left({7^2}\right)\) \(=\) \(\displaystyle \dfrac {7^3 - 1} {7 - 1}\) $\quad$ Sigma Function of Power of Prime $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {343 - 1} 6\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 57\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 3 \times 19\) $\quad$ $\quad$


\(\displaystyle \sigma \left({11^2}\right)\) \(=\) \(\displaystyle \dfrac {11^3 - 1} {11 - 1}\) $\quad$ Sigma Function of Power of Prime $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {1331 - 1} {10}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 133\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 7 \times 19\) $\quad$ $\quad$


\(\displaystyle \sigma \left({13}\right)\) \(=\) \(\displaystyle 14\) $\quad$ Sigma Function of Prime Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 7\) $\quad$ $\quad$


\(\displaystyle \sigma \left({17}\right)\) \(=\) \(\displaystyle 18\) $\quad$ Sigma Function of Prime Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 3^2\) $\quad$ $\quad$


\(\displaystyle \sigma \left({19^3}\right)\) \(=\) \(\displaystyle \dfrac {19^4 - 1} {19 - 1}\) $\quad$ Sigma Function of Power of Prime $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {130 \, 321 - 1} {18}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 7240\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^3 \times 5 \times 181\) $\quad$ $\quad$


\(\displaystyle \sigma \left({23}\right)\) \(=\) \(\displaystyle 24\) $\quad$ Sigma Function of Prime Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^3 \times 3\) $\quad$ $\quad$


\(\displaystyle \sigma \left({37}\right)\) \(=\) \(\displaystyle 38\) $\quad$ Sigma Function of Prime Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 19\) $\quad$ $\quad$


\(\displaystyle \sigma \left({181}\right)\) \(=\) \(\displaystyle 182\) $\quad$ Sigma Function of Prime Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 7 \times 13\) $\quad$ $\quad$


This gives a common factor of both $\sigma$ values of:

\(\displaystyle \) \(\) \(\displaystyle 6 \times 57 \times 133 \times 14 \times 18 \times 7240 \times 24 \times 38 \times 182\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^{11} \times 3^5 \times 5 \times 7^3 \times 13 \times 19^3\) $\quad$ $\quad$


The remaining prime factors of $m_1$:

\(\displaystyle 101 \times 8643 \times 1 \, 947 \, 938 \, 229\) \(=\) \(\displaystyle 101 \times \left({3 \times 43 \times 67}\right) \times \left({3^2 \times 739 \times 292 \, 879}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 3^3 \times 43 \times 67 \times 101 \times 739 \times 292 \, 879\) $\quad$ $\quad$


Thus:

\(\displaystyle \sigma \left({3^3}\right)\) \(=\) \(\displaystyle \dfrac {3^4 - 1} {3 - 1}\) $\quad$ Sigma Function of Power of Prime $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {81 - 1} 2\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 40\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^3 \times 5\) $\quad$ $\quad$


\(\displaystyle \sigma \left({43}\right)\) \(=\) \(\displaystyle 44\) $\quad$ Sigma Function of Prime Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^2 \times 11\) $\quad$ $\quad$


\(\displaystyle \sigma \left({67}\right)\) \(=\) \(\displaystyle 68\) $\quad$ Sigma Function of Prime Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^2 \times 17\) $\quad$ $\quad$


\(\displaystyle \sigma \left({101}\right)\) \(=\) \(\displaystyle 102\) $\quad$ Sigma Function of Prime Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 3 \times 17\) $\quad$ $\quad$


\(\displaystyle \sigma \left({739}\right)\) \(=\) \(\displaystyle 740\) $\quad$ Sigma Function of Prime Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^2 \times 5 \times 37\) $\quad$ $\quad$


\(\displaystyle \sigma \left({292 \, 879}\right)\) \(=\) \(\displaystyle 292 \, 880\) $\quad$ Sigma Function of Prime Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^4 \times 5 \times 7 \times 523\) $\quad$ $\quad$


This gives us the prime decomposition of the rest of $\sigma \left({m_1}\right)$:

\(\displaystyle \) \(\) \(\displaystyle 40 \times 44 \times 68 \times 102 \times 740 \times 292 \, 880\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({2^3 \times 5}\right) \times \left({2^2 \times 11}\right) \times \left({2^2 \times 17}\right) \times \left({2 \times 3 \times 17}\right) \times \left({2^2 \times 5 \times 37}\right) \times \left({2^4 \times 5 \times 7 \times 523}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^{14} \times 3 \times 5^3 \times 7 \times 11 \times 17^2 \times 37 \times 523\) $\quad$ $\quad$


The remaining prime factors of $m_2$:

\(\displaystyle 365 \, 147 \times 47 \, 303 \, 071 \, 129\) \(=\) \(\displaystyle 67 \times 127 \times 365 \, 147 \times 5 \, 559 \, 181\) $\quad$ $\quad$


Thus:

\(\displaystyle \sigma \left({67}\right)\) \(=\) \(\displaystyle 68\) $\quad$ Sigma Function of Prime Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^2 \times 17\) $\quad$ $\quad$


\(\displaystyle \sigma \left({127}\right)\) \(=\) \(\displaystyle 128\) $\quad$ Sigma Function of Prime Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^7\) $\quad$ $\quad$


\(\displaystyle \sigma \left({365 \, 147}\right)\) \(=\) \(\displaystyle 365 \, 148\) $\quad$ Sigma Function of Prime Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^2 \times 3^4 \times 7^2 \times 23\) $\quad$ $\quad$


\(\displaystyle \sigma \left({5 \, 559 \, 181}\right)\) \(=\) \(\displaystyle 5 \, 559 \, 182\) $\quad$ Sigma Function of Prime Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 173 \times 16 \, 067\) $\quad$ $\quad$


This gives us the prime decomposition of the rest of $\sigma \left({m_2}\right)$:

\(\displaystyle \) \(\) \(\displaystyle 68 \times 128 \times 365 \, 148 \times 292 \, 5 \, 559 \, 182\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({2^2 \times 17}\right) \times 2^7 \times \left({2^2 \times 3^4 \times 7^2 \times 23}\right) \times \left({2 \times 173 \times 16 \, 067}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^{12} \times 3^4 \times 7^2 \times 17 \times 23 \times 173 \times 16 \, 067\) $\quad$ $\quad$



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