The New Thoughts of Galileo

follow, 18416, 17296, and 9437056, 4363584. Now I put here the method that an excellent Geometer has given, to find an infinity of numbers similar to the preceding ones, that is to say, which being taken two by two, one is equal to the aliquot parts of the other, and reciprocally the other is equal to the aliquot parts of the first. Here is the rule.

If one takes the binary number two, or such other number as one would want, produced by the multiplication of the binary, provided that it be such that if one removes unity from the number which is triple to it, it is a prime number a number divisible only by one and itself; in the same way that the sextuple number, from which one removes unity, is a prime number: and finally, if unity being removed from the octodecuple eighteen-fold number of its square number multiplied by itself, it is still a prime number, and that one multiplies this last number by the double of the number one has taken, one will have a number whose aliquot parts will give another number, from which the aliquot parts will produce the preceding number: for example, I take three numbers, 2, 8, and 64, and find the three couples of the preceding numbers.