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**
H.P.Williams**

The Skewes family (with
variant spellings) is well documented in the book Cornish Heritage^{1 } 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 18^{th} 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
Transvaal in

Professor
Stanley Skewes

After a degree in Civil Engineering
at the ^{ 2} and 1955 ^{3 }.
Although he returned to

Among his contemporaries at King’s College, *Penguin Dictionary of
Curious and Interesting Numbers* the famous mathematician G.H.Hardy
described the Skewes number as ‘the largest number that has ever served any
definite purpose in mathematics‘. It is _{}. This number is truly huge. It could not be written down, in
usual notation, using all the books in the world. By comparison, the number of
particles in the universe is estimated at only 10^{80}, a minute fraction of the
size of Skewes’ Number. The number is also discussed in the book by the science
fiction writer Isaac Asimov, *Skewered!,
Of Matters Great and Small* ^{1 }. It and Stanley Skewes are also
cited in the 20^{th} edition of the Guinness Book of Records in 1973.
Among other honours he was elected a Fellow of the Royal Astronomical Society
as a result of his other interest in Astronomy.

**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:

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,…

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

** References **

1.Isaac Asimov*, Skewered!, Of Matters Great and Small*,
Ace Books, New

2._{}(x) – Li (x), *Journal of
the *

* Mathematical
Society, ***8**,
227-283,1933.

3._{}(x) – Li (x) _{} II, *Proceedings
of the*

* *

4.Keith Skues, Cornish
Heritage, Werner Shaw,

5.David Welles, The *Penguin Dictionary of Curious and
Interesting Numbers,*

* *Penguin Books,

**Acknowledgements**

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

**The Author**

Paul Williams is a professor
at The London School of Economics. He was born and brought up in