We call $n$ a spoof odd perfect number if $n$ is odd and and $n=km$ for two integers $k, m > 1$ such that $\sigma(k)(m + 1) = 2n$, where $\sigma$ is the sum-of-divisors function.

In a letter to Mersenne dated November $15$, $1638$, Descartes showed that $$d = {{3}^2}\cdot{{7}^2}\cdot{{11}^2}\cdot{{13}^2}\cdot{22021} = 198585576189$$ would be an odd perfect number if $22021$ were prime.

Here is my question:

How did Descartes come up with the spoof odd perfect number $198585576189$?

  • 2
    $\begingroup$ Spoof odd perfect numbers are also known as Descartes Numbers. $\endgroup$ – Mr Pie Mar 15 '18 at 14:57
  • $\begingroup$ Thanks @user477343. I do am aware of Banks, et. al's paper on the topic. =) $\endgroup$ – Jose Arnaldo Bebita-Dris Mar 19 '18 at 3:37

If $22021$ were prime, we would have $$\sigma(d)=\sigma(3^2\cdot 7^2\cdot 11^2\cdot 13^2)\cdot 22022=(1+3+3^2)\cdot(1+7+7^2)\cdot(1+11+11^2)\cdot(1+13+13^2)\cdot 22022=2d$$ which can be verified by multiplication

I guess Descartes calculated $\sigma(3^2\cdot 7^2\cdot 11^2\cdot 13^2)=3^2\cdot 7\cdot 13\cdot 19^2\cdot 61$ , tried to multiply with the $19^2\cdot 61$ not fitting to the $3^2\cdot 7^2\cdot 11^2\cdot 13^2$-part and was lucky.


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