By Carmichael R.D.
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The proof is interesting — though easy — and I’ll ask your help, Paulo. I don’t know if you “believe” that Part 1 is true, but “make believe” that you “don’t believe”. A lot of beliefs at this point, but soon there will be a logical, irrefutable proof. Paulo obeyed and said: Paulo. OK, Papa Paulo, I don’t believe that Part 1 is true. P. That means that you believe there is a natural number greater than 1 which is not a product of primes. Paulo. Yes, I agree. P. Then there is the smallest natural number, which is greater than 1, but not a product of primes.
But it is true that Landon Clay, who always liked mathematics, did not become a professional mathematician, but instead became very wealthy. Like many people, he eventually returned to his ﬁrst love and through his Clay Institute has been oﬀering US$ 1 000 000 for the solution of each of seven outstanding problems. To tell the truth, the problems are so hard that Clay will not have to write a check during his lifetime. P. Oh look! Time has passed and I did not teach you a thing. Paulo. You did, I learned about primes, cows, and business.
To ﬁnd a formula for the sum of all factors of a number n. The formula has to be expressed in terms of the factorization of n as a product of primes. I’ll derive the formula for an arbitrary natural number n > 1 and at the same time work with a numerical example, say, the number 432. We start: n is the product of powers of distinct primes. Let p1 , . . , pr be the distinct prime factors of n; for each prime pi let ei ≥ 1 be such that pei i divides n, but pei i +1 does not divide n, so n = pe11 × · · · × perr .
Theory of numbers by Carmichael R.D.