A generalised Fermat Prime is a prime number of the form for a >0. It is called ‘generalised’ as a Fermat Prime is a number of this form with a = 0.

The discovery was made by Sylvanus A. Zimmerman of the United States.

“Until now only 392 generalised Fermat primes had been found: this new discovery makes 393. At 6,253,210 digits long, it’s now the 12th largest of all known primes, and the second-largest known non-Mersenne prime.”

Recently, James Davis found a counterexample to John H. Conway’s ‘Climb to a Prime’ conjecture, for which Conway was offering $1,000 for a solution.

The conjecture states the following:

“Let n be a positive integer. Write the prime factorisation in the usual way, where the primes are written in ascending order and exponents of 1 are omitted. Then bring the exponents down to the line, omit the multiplication signs, giving a number f(n). Now repeat.”

For example, f(60) = f(2^2 x 3 x 5) = 2235. As 2235 = 3 x 5 x 149, f(2235) = 35149. Since 35149 is prime, we stop there.

Davis had a feeling that the counterexample would be of the form

where p is the largest prime factor of n. This motivated him to look for x of the form

The number Davis found was 13532385396179 = 13 x 53^2 x 3853 x 96179, which maps to itself under f (i.e. its a fixed point). So, f will never map this composite number to a prime, hence disproving the conjecture.

The Ulam Spiral, discovered in 1963 by Stanislaw Ulam, is a graphical depiction of the set of prime numbers.

If you were to arrange the positive numbers in a spiral, starting with one at the centre, then circle all of the prime numbers, what would you get? As prime numbers don’t have a predictive structure, you would expect to get little or even nothing out of arranging the primes this way. But, Ulam discovered something incredible:

Ulam Spiral

To his surprise, the circled numbers tended to line up along diagonal lines. In the 200×200 Ulam spiral shown above, diagonal lines are clearly visible, confirming the pattern. Although less prominent, horizontal and vertical lines can also be seen.

Even more amazing, this pattern still appears even if we don’t start with 1 at the centre!

There are many patterns on this plot. One of the simplest ones is that there are many integer constants b and c such that the function:

generates, a number of primes that is large by comparison with the proportion of primes among numbers of similar magnitude, as n counts up {1, 2, 3, …}.

A Leyland number is an integer of the form , where x and y are integers greater than 1. This condition is very important as, without it, every positive integer would be a Leyland number of the form x^{1} + 1^{x}.

They are named after Paul Leyland, a British number theorist who studied the factorisation of integers and primality testing.

Leyland numbers are of interest as some of them are very large primes.

Leyland Primes

A Leyland prime is a Leyland number that is also prime. The first of such primes are:

Today I thought I’d quickly discuss a extremely important theorem in one of my favourite areas in mathematics: Number Theory (as you can probably tell by the number of posts that I’ve published about primes!).

Perhaps the first property of π(x) – the number of primes less than or equal to x – is that π(x) tends to infinity as x tends to infinity. In other words, the prime numbers are infinite, which was proved by Euclid in “Elements”. A more precise result, established by Euler in 1737, was that the series of reciprocals of the prime numbers:

is a divergent series. In doing so, Euler found an alternative way to prove that there was an infinite number of primes, as if there wasn’t then the series would have a finite value.

The Prime Number Theorem states that if π(x) is the number of primes less than or equal to x, then

Although the notation ~ may be unfamiliar, it simply means that π(x) is asymptotically equal to x/lnx, i.e.

Note that the prime number theorem is equivalent to saying that the nth prime number p_{n} satisfies the following relationship:

The PNT was proposed by Gauss in 1792 when he was only 15 years old! (Makes you wonder what you’ve been doing with your life so far…) He later refined this estimate to

In number theory, Mills’ theorem states that there exists a real constant A such that is prime for all positive integers n (note that this is a floor function). Mills’ constant is defined as the smallest real positive number such that Mills’ theorem is true.

This constant is named after William H. Mills, who proved the existence of A based on results of Guido Hoheisel and Albert Ingham on the prime gaps in 1947.

The primes generated using Mills’ constant are known as Mills’ primes:

“If a_{i} denotes the i^{th} prime in this sequence, then a_{i} can be calculated as the smallest prime number larger than . In order to ensure that rounding produces this sequence of primes, it must be the case that .”

In 2005, Caldwell and Cheng computed 6850 base 10 digits of Mills’ constant under the assumption that the Riemann hypothesis is true.

Hope you enjoyed this installment of ‘Math’s Bite’! M x

where π(x) is the prime-counting function and li(x) is logarithmic integral function.

Prime Counting Function: is the function counting the number of prime numbers less than or equal to some real number x.

Logarithmic Integral Function: used in prime number theorem as an estimate of the number of prime numbers less than a given value.

In 1912, it was proved by Littlewood that this limit existed, and it – Skewes’ number – was subsequently found by Stanley Skewes, South African mathematician, in 1933.

However this limit has been improved to by Lehman in 1966.