Amicable Pair with Smallest Common Prime Factor 5/Mistake

Source Work

 * The Dictionary
 * $220$
 * $220$

Mistake

 * Most known amicable pairs have both numbers in the pair divisible by $3$. However, this is not a general rule: this counterexample by te Riele may be the smallest such: $5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181$ multiplied by either $101 \times 8643 \times 1 \, 947 \, 938 \, 229$ or by $365 \, 147 \times 47 \, 303 \, 071 \, 129$.

Correction
Those numbers are incorrect.

This should read:
 * ... multiplied by either $101 \times 8693 \times 19 \, 479 \, 382 \, 229$ or by $365 \, 147 \times 47 \, 307 \, 071 \, 129$.

Analysing the $\sigma$ values of the numbers given reveals the incorrectitude.

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

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

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

First we make sure we have all the prime factors:

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:

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

The remaining prime factors of $m_1$:

Thus:

This gives us the prime decomposition of the rest of $\map \sigma {m_1}$:

The remaining prime factors of $m_2$:

Thus:

This gives us the prime decomposition of the rest of $\map \sigma {m_2}$:

Thus it is seen that $\map \sigma {m_1} \ne \map \sigma {m_2}$, and so $m_1$ and $m_2$ are not amicable.

The error originates in.

He corrects this mistake in :


 * In an $87-05-15$ letter announced a $33$-digit specimen (misquoted in UPINT2)


 * $5 \cdot 7^2 \cdot 11^2 \cdot 13 \cdot 17 \cdot 19^3 \cdot 23 \cdot 37 \cdot 181 \begin{cases} 101 \cdot 8693 \cdot 19479382229 \\ 365147 \cdot 47307071129 \end{cases}$

Smaller counterexamples have since been found.