The Skewes family (with variant spellings) is well documented in the book Cornish Heritage1 by Keith Skues. Many originate from Cury on the Lizard peninsula. A number of farms on the Lizard, and close by, bear the name ‘Skewes’ and many members of the family still live in Cornwall as well as there being descendants overseas whose ancestors emigrated to metal mining regions in South and North America, Mexico, Australia and South Africa. Keith Skues remarks “ The family may not have produced any kings, queens, prime ministers, statesmen or internationally known figures. It is a typical middle/working class family.“. There are many interesting aspects to the places and family, from the famous siege of Skewes which took place at the Skewes farmhouse near Nancegollan in the 18th century to ‘Froggie’ Skewes who, within the author’s memory, bred frogs for commercial purposes on Carn Brea.
What is not widely known in
It was discovered by Stanley
Skewes who was the son of Henry and Emily Skewes. They emigrated to the
Professor Stanley Skewes
After a degree in Civil Engineering
Among his contemporaries at King’s College,
The Skewes Number
The number arises from studying the prime numbers. These are numbers which are not divisible by any number other than 1 and themselves (0 and 1 are not regarded as prime numbers). They are sometimes referred to as the ‘building blocks of arithmetic’ since any number can be factored uniquely into primes.. The first few prime numbers are:
Many attempts have been made to find a ‘formula’ which will generate them but one has never been found. Although they are irregular it has long been recognised that they get less dense as one gets to larger and larger numbers. A famous result (the ‘Prime Number Theorem’ due to Gauss) shows that around any number x the density is about 1/log x where log x is the natural logarithm of x (the logarithm to the base e = 2.718...). So at 100 one would expect about 1 in 5 numbers to be prime but at 1000 one would expect only about 1 in 7 to be prime. This estimate allows one to work out roughly how many prime numbers there are less than any given number. One has to sum up all the densities up to the given number n. Technically this is done by what is known as an integral . The value of this expression can be calculated for different values of n and is given in the table below together with the actual number of prime numbers up to n.
n Number of Primes
less than n
1000 168 178
10000 1229 1246
50000 5133 5167
100000 9592 9630
500000 41538 41606
1000000 78498 78628
2000000 148933 149055
5000000 348513 348638
10000000 664579 664918
20000000 1270607 1270905
90000000 5216954 5217810
100000000 5761455 5762209
1000000000 50847534 50849235
10000000000 455052511 455055614
It can be seen that the integral gives a remarkably close approximation to the number of primes as n gets large. In fact, the ratio of the integral to the actual number approaches 1 as n approaches infinity.
Notice, however, that the estimate is always an overestimate. It remains an overestimate for astronomically large values of n. It was thought for a long time that this would always be the case until it was proved by Littlewood that it would eventually become an underestimate (but then switch back again and alternate between an underestimate and overestimate an infinite number of times). Skewes showed that it must switch to being an underestimate before n reaches . This is the Skewes Number. In fact, his proof depends on assuming that one of the most famous conjectures of mathematics, the Riemann Hypothesis, is true. If the Riemann Hypothesis were not true then it would still become an underestimate before some even larger number, sometimes known as Skewes Second Number. This is .
This demonstrates, among other things, the fallacy of assuming that because something is true a very large (in this case astronomical) number of times it will always be true!
Since Skewes proved these results his number has been reduced in size but remains known as the Skewes Number.
A memorandum, written by Skewes on his retirement, which discusses the Skewes Number and further developments is kept in a glass case in the Mathematics Department of Capetown University where he worked as a lecturer then a professor. He died in 1988 at the age of 89 .
Skewes Family Bible
1.Isaac Asimov, Skewered!, Of Matters Great and Small, Ace Books, New
Mathematical Society, 8, 227-283,1933.
4.Keith Skues, Cornish
Heritage, Werner Shaw,
5.David Welles, The Penguin Dictionary of Curious and Interesting Numbers,
The author would like to
acknowledge Professor Brian Griffiths who first brought the Skewes Number to
his attention, and the help of Dr Kenneth Hughes of
Paul Williams is a professor
at The London School of Economics. He was born and brought up in