Amicable Pair with Smallest Common Prime Factor 5

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Theorem

A amicable pair whose smallest common prime factor is greater than $3$ has the elements:

$m_1 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 101 \times 8693 \times 19 \, 479 \, 382 \, 229$

and:

$m_2 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 365 \, 147 \times 47 \, 307 \, 071 \, 129$


This was at one point the smallest known counterexample to the observation that:

most amicable pairs 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:

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

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 $\dfrac {p_i^2 - 1} {p_i - 1} = \dfrac {\paren {p_i + 1} \paren {p_i - 1} } {p_i - 1} = p_i + 1$.


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:

\(\displaystyle \map \sigma 5\) \(=\) \(\displaystyle 6\) $\quad$ Sigma Function of Prime Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 3\) $\quad$ $\quad$


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


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


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


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


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


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


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


\(\displaystyle \map \sigma {181}\) \(=\) \(\displaystyle 182\) $\quad$ 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:

\(\displaystyle \) \(\) \(\displaystyle 6 \times 57 \times 133 \times 14 \times 18 \times 7240 \times 24 \times 38 \times 182\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^{11} \times 3^5 \times 5 \times 7^3 \times 13 \times 19^3 \times 181\) $\quad$ $\quad$


The remaining prime factors of $m_1$:

\(\displaystyle \map \sigma {101 \times 8693 \times 19 \, 479 \, 382 \, 229}\) \(=\) \(\displaystyle \paren {101 + 1} \paren {8693 + 1} \paren {19 \, 479 \, 382 \, 229 + 1}\) $\quad$ Sigma Function of Square-Free Integer $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \paren {2 \times 3 \times 17} \paren {2 \times 3^3 \times 7 \times 23} \paren {2 \times 3 \times 5 \times 7 \times 11^2 \times 37 \times 20 \, 719}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^3 \times 3^5 \times 5 \times 7^2 \times 11^2 \times 17 \times 23 \times 37 \times 20 \, 719\) $\quad$ $\quad$


The remaining prime factors of $m_2$:

\(\displaystyle \map \sigma {365 \, 147 \times 47 \, 307 \, 071 \, 129}\) \(=\) \(\displaystyle \paren {365 \, 147 + 1} \paren {47 \, 307 \, 071 \, 129 + 1}\) $\quad$ Sigma Function of Square-Free Integer $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \paren {2^2 \times 3^4 \times 7^2 \times 23} \paren {2 \times 3 \times 5 \times 11^2 \times 17 \times 37 \times 20 \, 719}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^3 \times 3^5 \times 5 \times 7^2 \times 11^2 \times 17 \times 23 \times 37 \times 20 \, 719\) $\quad$ $\quad$


and so:

\(\displaystyle \map \sigma {m_1} = \map \sigma {m_2}\) \(=\) \(\displaystyle \paren {2^{11} \times 3^5 \times 5 \times 7^3 \times 13 \times 19^3 \times 181} \paren {2^3 \times 3^5 \times 5 \times 7^2 \times 11^2 \times 17 \times 23 \times 37 \times 20 \, 719}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^{14} \times 3^{10} \times 5^2 \times 7^5 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 20 \, 719\) $\quad$ $\quad$


It remains to be shown that:

$\map \sigma {m_1} = \map \sigma {m_2} = m_1 + m_2$

So:

\(\displaystyle m_1 + m_2\) \(=\) \(\displaystyle 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times \paren {101 \times 8693 \times 19 \, 479 \, 382 \, 229 + 365 \, 147 \times 47 \, 307 \, 071 \, 129}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 6921698310331805 \times \paren {17102761241386397 + 17274035101540963}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 6921698310331805 \times 34376796342927360\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 237945813161460881075326612684800\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^{14} \times 3^{10} \times 5^2 \times 7^5 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 20719\) $\quad$ $\quad$

$\blacksquare$


Historical Note

In $1986$, David Wells reported in his Curious and Interesting Numbers of $1986$ that "numerous mathematicians" had conjectured that all amicable pairs both of whose elements are odd are multiples of $3$.

This amicable pair, a counterexample, was communicated by letter dated $15$th May $1987$ from Herman te Riele to Richard K. Guy.

The latter published it, with mistakes, in his Unsolved Problems in Number Theory, 2nd ed. of $1994$:

$5 \cdot 7^2 \cdot 11^2 \cdot 13 \cdot 17 \cdot 19^3 \cdot 23 \cdot 37 \cdot 181 \begin{cases} 101 \cdot 8643 \cdot 1947938229 \\ 365147 \cdot 47303071129 \end{cases}$


David Wells repeated the mistakes in his Curious and Interesting Numbers, 2nd ed. of $1997$.

Guy finally published the corrected version in his Unsolved Problems in Number Theory, 3rd ed. of $2004$.


It was for some time believed to be the smallest such amicable pair, but smaller ones have since been discovered.


Sources