BOUGEAREL Alain wrote:Nicomaque : Livre I chapitre XI paragraphes 2 et 3

...You find it i Nicomacus. ll give you the references in a little moment''...

The prime and incomposite

P. 202

https://ia600709.us.archive.org/27/item ... hmetic.pdf
Extrait :

*Now the first species, the prime and incomposite, is found when an odd number admits of no other factor save the one woth the number itself as denominator which is always unity 1; for example,*

3,7,11,13,17,19,23,29,31
Thus 31 is the Tenth prime and composite

The relationship I mentioned, between perfect and prime numbers (1+2+4+8+16=31, a prime number, which multiplied by the last term results in a perfect number), is on p.210/211:

"You must set forth the even-times even numbers from unity, advancing

in order in one line, as far as you please: I, 2,4,8, 16,32,64,

128, 256, 5 I 2, 1,024, 2,048, 4,096. . .. Then you must add them together,

one at a time, and each time you make a summation observe

the result to see what it is. If you find that it is a prime, incomposite

number, multiply it by the quantity of the last number added, and

the result will always be a perfect number. If, however, the result is

secondary and composite, do not multiply, but add the next and observe

again what the resulting number is, if it is secondary and composite,

again pass it by and do not multiply; but add the next; but

if it is prime and incomposite, multiply it by the last term added, and

the result will be a perfect number; and so on to infinity. In similar

fashion you will produce all the perfect numbers in succession, overlooking

none....

...When these have been discovered, 6 among the units and 28 in the 6

tens, you must do the same to fashion the next. Again add the next 7

number, 8, and the sum is 15. Observing this, I find that we no longer

have a prime and incomposite number, but in addition to the factor

with denominator like the number itself,' it has also a fifth and a

third, with unlike denominators. Hence I do not multiply it by 8,

but add the next number, 16, and 31 results. As this is a prime, incomposite

number, of necessity it will be multiplied, in accordance

with the general rule of the process, by the last number added, 16, and

the result is 496, in the hundreds; and then comes 8,128 in the thousands,

and so on, as far as it is convenient for one to follow."

The proposition for this is first found in Euclid, and obviously may subsequently found in books on arithmetic (neo-pythagorean or otherwise).