The unknown
Outstanding unknown areas concerning primes are the ‘Twin primes problem’
and the famous ‘Goldbach conjecture’.
Twin primes are pairs of consecutive primes separated only by an even
number. The twin primes in the range from 1 to 100 are 3, 5; 5, 7; 11, 13; 17,
19; 29, 31; 41, 43; 59, 61; 71, 73. On the numerical front, it is known there are
27,412,679 twins less than 10^10. This means the even numbers with twins, like
12 (having twins 11, 13), constitute only 0.274% of the numbers in this range.
Are there an infinite number of twin primes? It would be curious if there were
not, but no one has so far been able write down a proof of this.
Christian Goldbach conjectured that:
Every even number greater than 2 is the sum of two prime numbers.
The number of the numerologist
One of the most challenging areas of number theory concerns ‘Waring’s problem’. In 1770
Edward Waring, a professor at Cambridge, posed problems involving writing whole numbers as the
addition of powers. In this setting the magic arts of numerology meet the clinical science of
mathematics in the shape of primes, sums of squares and sums of cubes. In numerology, take the
unrivalled cult number 666, the ‘number of the beast’ in the biblical book of Revelation, and which
has some unexpected properties. It is the sum of the squares of the first 7 primes:
666 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2
Numerologists will also be keen to point out that it is the sum of palindromic cubes and, if that is
not enough, the keystone 6^3 in the centre is shorthand for 6 × 6 × 6:
666 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 5^3 + 4^3 + 3^3 + 2^3 + 1^3
The number 666 is truly the ‘number of the numerologist’.
For instance, 42 is an even number and we can write it as 5 + 37. The fact
that we can also write it as 11 + 31, 13 + 29 or 19 + 23 is beside the point – all
we need is one way. The conjecture is true for a huge range of numbers – but it
has never been proved in general. However, progress has been made, and some
have a feeling that a proof is not far off. The Chinese mathematician Chen
Jingrun made a great step. His theorem states that every sufficiently large even
number can be written as the sum of two primes or the sum of a prime and a
semi-prime (a number which is the multiplication of two primes).
The great number theorist Pierre de Fermat proved that primes of the form 4k