LETTERS TO THE EDITORS
To the Editors:
I have two remarks regarding Sarah Glaz’s interesting column “Ode to Prime Numbers” (Macroscope, July–August).
First, it is true that Leonhard Euler was the first to study the zeta function in depth, as Glaz says, but it wasn’t “in the early 1800s”—the great mathematician died in 1783 (a prime number!). Second, contrary to the author’s statement, there are ingenious formulas that yield the full set of prime numbers. On the other hand, as far as I know, none is even close to being computationally practical. Here is one that has been called “prime beef,” based on Wilson’s Theorem: Let m and n be nonnegative integers, and let k = n! – m(n + 1) + 1. Then, f(m, n) = 2 + ½ (n – 1) [|k2 – 1| – (k2 – 1)] yields all and only primes. Don’t be surprised if the equation yields only 2 for a long time as you move through increasing integers. But eventually, for example, f(329,891, 10) = 11. I suppose even a computer would go batty calculating primes this way.
Luis F. Moreno
Dr. Glaz responds:
Euler indeed died in 1783. He gave the infinite product formula in 1737—“early in the 18th century” and not “in the early 1800s.” I am happy for the opportunity to correct this typo and also thank Michael Rochester (Professor Emeritus at Memorial University) for letting me know of it in a private email.
On the other hand, my statement about prime numbers, “no formula has been found that covers them all,” is correct. There exist a number of formulas supposed to generate all the prime numbers, but in practice all of them are deficient in some way. An informative essay about formulas for primes may be found at Wolfram Math World (http://mathworld.wolfram.com/PrimeFormulas.html). The essay begins as follows: “There exist a variety of formulas for either producing the nth prime as a function of n or taking on only prime values. However, all such formulas require either extremely accurate knowledge of some unknown constant, or effectively require knowledge of the primes ahead of time in order to use the formula.” Although formulas for primes seem to exist, for all practical purposes, they do not. I hope the situation will be remedied in the future. Meanwhile, it is an interesting topic for a poem.