Fermat's Library | Perfect Numbers annotated/explained version.
source link: https://fermatslibrary.com/s/perfect-numbers
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If you want to run this program yourself, David Roberts and Christopher Long created a repo on Github with the code: https://github.com/DavidMichaelRoberts/Tao_1983/blob/master/perfect.bas
They changed the 'up arrow' notation ↑ in the original code to the modern caret ^. If you want to run the this code yourself, Christopher Long uploaded it to Github: https://github.com/DavidMichaelRoberts/Tao_1983/blob/master/primes.bas Here is Euclid's proof that if $2^n−1$ is prime, then $N=2^{n−1} (2n−1)$ is perfect.
Proof: Clearly the only prime divisors of N are $2^n−1$ and 2. Since $2^n−1$ occurs as a single prime, we have simply that $σ(2^n−1)=(1+(2n−1))=2^n$, and thus $$ σ(N) = σ(2^{n−1})σ(2^n−1) = \frac{2^n−1}{2-1}2^n=2^n(2^n-1)=2N $$ So N is perfect!
The task of finding perfect numbers, then, is connected with finding primes of the form $2^n − 1$, also known as Mersenne primes. Mersenne primes were named after the seventeenth century monk Marin Mersenne, a colleague of Descartes, Fermat and Pascal. Many of the largest known primes are Mersenne primes. Additionally, if you want to learn more about Tao's extraordinary mathematical abilities at an early age there's a [really good video by Tibees](https://www.youtube.com/watch?v=I_IFTN2Toak). She explores in depth a report by a researcher of mathematically gifted children that studied Tao at the age of 7. This was Terence Tao's first paper. Tao, recipient of the Fields Medal in 2006 and one of the greatest living mathematicians submitted this paper to a students' math journal called Trigon at the age of 8.
In this paper, Tao developed a program in BASIC to find perfect numbers.
![](http://static01.nyt.com/images/2015/07/26/magazine/26tao2/26mag-26tao-t_CA0-articleLarge.jpg)
David Roberts and Christopher Long converted the original paper published in Trigon to this LaTeX version.
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