On Numbers: On Prime Numbers (-1499)

“A prime number is one (which is) measured by a unit alone.” (Euclid, “The Elements”, Book VII, cca 300 BC)

"If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect." (Euclid,"Elements", cca 300 BC)

“Numbers prime to one another are those which are measured by a unit alone as a common measure.” (Euclid, “The Elements”, Book VII, cca 300 BC)

"Two unequal numbers being set out, and the less being continually subtracted in tum from the greater, if the number which is left never measures the one before it until an unit is left, the original numbers will be prime to one another." (Euclid, Book VII, cca 300 BC)

"There exists an elegant and sure method of generating these numbers, which does not leave out any perfect numbers and which does not include any that are not; and which is done in the following way. First set out in order the powers of two in a line, starting from unity, and proceeding as far as you wish: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096; and then they must be totalled each time there is a new term, and at each totaling examine the result, if you find that it is prime and non-composite, you must multiply it by the quantity of the last term that you added to the line, and the product will always be perfect. If, otherwise, it is composite and not prime, do not multiply it, but add on the next term, and again examine the result, and if it is composite leave it aside, without multiplying it, and add on the next term. If, on the other hand, it is prime, and non-composite, you must multiply it by the last term taken for its composition, and the number that results will be perfect, and so on as far as infinity." (Nicomachus of Gerasa,"Introductio Arithmetica", cca. 100 AD)

"Numbers are called prime which can be divided by no number; they are seen to be not ‘divisible’ by the monad but ‘composed’ of it: take, for example, the numbers live, seven, eleven, thirteen, seventeen, and others like them. No number can divide these numbers into integers. So, they are called `prime,' since they arise from no number and are not divisible into equal proportions. Arising in themselves, they beget other numbers from themselves, since even numbers are begotten from odd numbers, but an odd number cannot be begotten from even numbers. Therefore, prime numbers must of necessity be regarded as beautiful." (Martianus Capella, cca. 400 AD)

"Number is divided into even and odd. Even number is divided into the following: evenly even, evenly uneven, and unevenly uneven. Odd number is divided into the following: prime and incomposite, composite, and a third intermediate class" (mediocris) which in a certain way is prime and incomposite but in another way secondary and composite." (Isidore of Seville, Etymologies, Book III, cca. 600