Four Personalized Prime Number Formulae

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The recent article, “Complex Tale of Eisenstein Prime Numbers“, was devoted to the prime numbers found by Ferdinand Gotthold Max Eisenstein.

"Pierre de Fermat" by Magnus Manske

Pierre de Fermat: Image by Magnus Manske

The names of several other mathematicians have become associated with their own sets of primes numbers. This article will introduce some of these very personalized primes.

Fermat Numbers and Fermat Primes

Pierre de Fermat is perhaps best known for “Fermat’s Last Theorem”. Here, however, the focus is on Fermat numbers and Fermat primes.

One calculates a Fermat number as a function of the natural number ‘n’, where ‘n’ is greater than or equal to zero. The formula for a Fermat number is F[n] = 2^(2^n) + 1.
Fermat numbers become very large very quickly. To the limits of current computations, only the first five Fermat numbers are primes. In 1650, Fermat had conjectured that each and every Fermat number would be prime. This conjecture is false.

"Fermat Numbers" by Mike DeHaan

Fermat Numbers: Chart by Mike DeHaan

Much later, our old friend Eisenstein posed a problem: to prove that there are infinitely many Fermat prime numbers. This problem has neither been proven nor disproven.

Gauss found an interesting geometrical use for Fermat prime numbers. The problem is to subdivide a circle, using only a straightedge and a compass. (That’s the compass that is the circle-drawing mathematician’s tool, not the magnetized needle used in orienteering to find the North magnetic pole).

Gauss proved that one may subdivide a circle into ‘n’ equal parts, if and only if n = 2^k * F[i] * F[j] where F[i] and F[j] are different Fermat primes and ‘k’ is a natural number.

Gaussian Integers and Gaussian Primes

Named for German mathematician Carl Friedrich Gauss, the Gaussian integers are complex numbers, much like Eisenstein numbers, of the form z = a + b*i where ‘z’ is complex, ‘a’ and ‘b’ are integers, and ‘i’ is the square root of negative one.

A composite Gaussian integer is one that is the product of two other Gaussian integers; much as a composite natural number is the product formed by multiplying two smaller natural numbers together.

A Gaussian number, z = a + b*i, has three distinct ways to qualify as a Gaussian prime.

  • If and only if (a^2 + b^2) is a prime in the natural numbers, and neither ‘a’ nor ‘b’ are equal to zero, then ‘z’ is a Gaussian prime.
  • For z = 0 + b*i, if and only if the absolute value |b| is prime and |b| is equivalent to 3 modulo 4, then ‘z’ is a Gaussian prime.
  • For z = a + 0*i, if and only if the absolute value |a| is prime and |a| is equivalent to 3 modulo 4, then ‘z’ is a Gaussian prime.

The phrase “|n| is equivalent to 3 modulo 4″ means that “n/4 has a remainder of 3″. For example, the first few natural numbers that are also Gaussian primes are 3, 7 and 11.

Sophie Germain Primes Rely on Others

The French mathematician Sophie Germain defined and studied her namesake prime numbers.

"Sophie Germain" by El Bibliomata

Sophie Germain: Image by El Bibliomata

Any prime number ‘p’ qualifies as a Sophie Germain prime if both ‘p’ and (2*p + 1) are prime. The early examples are 2, 3, 5, 11 and 23.

Whether there are infinitely many Sophie Germain primes also remains an unsolved question. A report in 2010 states that the largest found to that date has 79,911 decimal digits!

Germain had contributed a very partial proof towards “Fermat’s Last Theorem”. His theorem states that for non-zero integers ‘a’, ‘b’, ‘c’ and ‘n’, with n>2, there is no solution for a^n + b^n = c^n.

Germain replaced the general power ‘n’ with her Sophie Germain prime, ‘p’, and also stipulated that neither ‘a’, ‘b’ nor ‘c’ could be a multiple of ‘p’. She then proved that there is no solution for a^p + b^p = c^p.

The Circular Reasoning of Pierpont Primes

James Pierpont, an American, defined his namesake primes as any prime formed by p = (2^i)*(3^j) + 1, where ‘i’ and ‘j’ are natural numbers. As with Fermat, many of Pierpont’s numbers are not prime.

"Pierpont Numbers" by Mike DeHaan

Pierpont Numbers: Chart by Mike DeHaan

Mathematicians suspect that there are infinitely many Pierpont primes, which begin as {2, 3, 5, 7, 13, 17…}.

Since ‘i’ and ‘j’ may be zero, the Fermat primes are included in the Pierpont primes where ‘i’ is 1, 2, 4, 9 and 16.

Pierpont extended a line of reasoning originally presented by Gauss. Rather than dividing a circle using a compass and straightedge, he “simply” folded the circle using origami techniques.

Pierpont proved that folding a circle into ‘n’ equal parts works if, and only if, ‘n’ is a Pierpont prime.

The Endless Possibilities for Personalized Primes

Other mathematicians have studied their own specific formulae for generating interesting sets of prime numbers. Three problems quickly arise. First, the formula may create composite numbers along with primes. Second, the formula skips some prime numbers. Finally, the formula may generate very large numbers fairly quickly in the process.

It seems there is ample room for anyone to create their own patterns of numbers and test them to find interesting or useful prime numbers that they can call their very own.

References:

Caldwell, Chris K. University of Tennessee at Martin. Pierpont Prime. (1999-2011). Accessed Nov. 17, 2011.

Weisstein, Eric W. MathWorld–A Wolfram Web Resource. Fermat PrimeGaussian PrimeSophie Germain Prime. (1999-2011). Accessed Nov. 17, 2011.

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© Copyright 2011 Mike DeHaan, All rights Reserved. Written For: Decoded Science
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  1. [...] Today’s belated article at Decoded Science shows how four mathematicians became entangled with each others’ personalized prime numbers. Learn the splendid details in “Four Personalized Prime Number Formulae“. [...]

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