Amicable Pair with Smallest Common Prime Factor 5/Mistake
Source Work
1997: David Wells: Curious and Interesting Numbers (2nd ed.):
- The Dictionary
- $220$
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 divisor sum 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 Divisor Sum of Integer:
- $\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where:
- $\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i} = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$
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:
\(\ds 8643\) | \(=\) | \(\ds 3 \times 43 \times 67\) | ||||||||||||
\(\ds 1 \, 947 \, 938 \, 229\) | \(=\) | \(\ds 3^2 \times 739 \times 292 \, 879\) | ||||||||||||
\(\ds 47 \, 303 \, 071 \, 129\) | \(=\) | \(\ds 67 \times 127 \times 5 \, 559 \, 181\) |
All other factors given are indeed prime.
We establish the contributions to the divisor sum 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:
\(\ds \map {\sigma_1} 5\) | \(=\) | \(\ds 6\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3\) |
\(\ds \map {\sigma_1} {7^2}\) | \(=\) | \(\ds \dfrac {7^3 - 1} {7 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {343 - 1} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 57\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 19\) |
\(\ds \map {\sigma_1} {11^2}\) | \(=\) | \(\ds \dfrac {11^3 - 1} {11 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {1331 - 1} {10}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 133\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 \times 19\) |
\(\ds \map {\sigma_1} {13}\) | \(=\) | \(\ds 14\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 7\) |
\(\ds \map {\sigma_1} {17}\) | \(=\) | \(\ds 18\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3^2\) |
\(\ds \map {\sigma_1} {19^3}\) | \(=\) | \(\ds \dfrac {19^4 - 1} {19 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {130 \, 321 - 1} {18}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7240\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^3 \times 5 \times 181\) |
\(\ds \map {\sigma_1} {23}\) | \(=\) | \(\ds 24\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^3 \times 3\) |
\(\ds \map {\sigma_1} {37}\) | \(=\) | \(\ds 38\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 19\) |
\(\ds \map {\sigma_1} {181}\) | \(=\) | \(\ds 182\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 7 \times 13\) |
This gives a common factor of both $\sigma_1$ values of:
\(\ds \) | \(\) | \(\ds 6 \times 57 \times 133 \times 14 \times 18 \times 7240 \times 24 \times 38 \times 182\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^{11} \times 3^5 \times 5 \times 7^3 \times 13 \times 19^3\) |
The remaining prime factors of $m_1$:
\(\ds 101 \times 8643 \times 1 \, 947 \, 938 \, 229\) | \(=\) | \(\ds 101 \times \paren {3 \times 43 \times 67} \times \paren {3^2 \times 739 \times 292 \, 879}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3^3 \times 43 \times 67 \times 101 \times 739 \times 292 \, 879\) |
Thus:
\(\ds \map {\sigma_1} {3^3}\) | \(=\) | \(\ds \dfrac {3^4 - 1} {3 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {81 - 1} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 40\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^3 \times 5\) |
\(\ds \map {\sigma_1} {43}\) | \(=\) | \(\ds 44\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 11\) |
\(\ds \map {\sigma_1} {67}\) | \(=\) | \(\ds 68\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 17\) |
\(\ds \map {\sigma_1} {101}\) | \(=\) | \(\ds 102\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3 \times 17\) |
\(\ds \map {\sigma_1} {739}\) | \(=\) | \(\ds 740\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 5 \times 37\) |
\(\ds \map {\sigma_1} {292 \, 879}\) | \(=\) | \(\ds 292 \, 880\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^4 \times 5 \times 7 \times 523\) |
This gives us the prime decomposition of the rest of $\map {\sigma_1} {m_1}$:
\(\ds \) | \(\) | \(\ds 40 \times 44 \times 68 \times 102 \times 740 \times 292 \, 880\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2^3 \times 5} \times \paren {2^2 \times 11} \times \paren {2^2 \times 17} \times \paren {2 \times 3 \times 17} \times \paren {2^2 \times 5 \times 37} \times \paren {2^4 \times 5 \times 7 \times 523}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^{14} \times 3 \times 5^3 \times 7 \times 11 \times 17^2 \times 37 \times 523\) |
The remaining prime factors of $m_2$:
\(\ds 365 \, 147 \times 47 \, 303 \, 071 \, 129\) | \(=\) | \(\ds 67 \times 127 \times 365 \, 147 \times 5 \, 559 \, 181\) |
Thus:
\(\ds \map {\sigma_1} {67}\) | \(=\) | \(\ds 68\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 17\) |
\(\ds \map {\sigma_1} {127}\) | \(=\) | \(\ds 128\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^7\) |
\(\ds \map {\sigma_1} {365 \, 147}\) | \(=\) | \(\ds 365 \, 148\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 3^4 \times 7^2 \times 23\) |
\(\ds \map {\sigma_1} {5 \, 559 \, 181}\) | \(=\) | \(\ds 5 \, 559 \, 182\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 173 \times 16 \, 067\) |
This gives us the prime decomposition of the rest of $\map {\sigma_1} {m_2}$:
\(\ds \) | \(\) | \(\ds 68 \times 128 \times 365 \, 148 \times 292 \, 5 \, 559 \, 182\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2^2 \times 17} \times 2^7 \times \paren {2^2 \times 3^4 \times 7^2 \times 23} \times \paren {2 \times 173 \times 16 \, 067}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^{12} \times 3^4 \times 7^2 \times 17 \times 23 \times 173 \times 16 \, 067\) |
Thus it is seen that $\map {\sigma_1} {m_1} \ne \map {\sigma_1} {m_2}$, and so $m_1$ and $m_2$ are not amicable.
$\blacksquare$
The error originates in 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.).
He corrects this mistake in 2004: Richard K. Guy: Unsolved Problems in Number Theory (3rd ed.):
- In an $87-05-15$ letter te Riele 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.
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $220$