Elementary arithmetic, with brief notices of its history... by Robert Potts.

4

Hence it follows that all prime numbers, except 2 and 5, will end
either with 1, 3, 7, or 9.
Next, the number 3 being a prime number, every fourth number
is divisible by 3, and may be omitted.
Similarly, the number 7 being a prime number, every eighth
number is divisible by 7, and may be omitted; and so on, omitting
all numbers divisible by each of the successive prime numbers in
order, until at length all the composite numbers are sifted out. Then
the remaining numbers will be the prime numbers contained in the
given series.
As an exemplification of the process, let it be required to find all
the prime numbers contained between the limits of 1 and 100.
Having written down in order the first 100 numbers; first, 2 and
and 5 are prime numbers; omitting 2 and 5, and all multiples of
them, there will be found 31 of the numbers left.
Next omitting 9, 21, 27, 33, 51, multiples of 3, a prime number;
also 49, 77, 91, multiples of 7, a prime number.
And the remaining numbers are11 19 31 43 59 71 83
13 23 37 47 61 73 89
17 29 41 53 67 79 97
and of these no one of the larger numbers is a multiple of any one
of the smaller prime numbers; it follows that these must be prime
numbers.
And there are 25 or 26 (if unity be included) prime numbers in
the first hundred of the natural numbers.'
1 The sieve of Eratosthenes was the name given to a contrivance he invented for
finding the prime numbers. His method consisted in writing the natural numbers
in order, beginning with unity, and continuing the series to any extent. He then
separated or sifted out all numbers that were not prime numbers, and by this means
he ascertained the prime numbers in the order of their magnitudes. The problem
of finding in general a prime number, beyond a certain limit, by a direct process, has
engaged the attention of mathematicians, but no results of a satisfactory nature have
yet been attained. No algebraical formula has yet been discovered which will
contain prime numbers only, as all known formula for prime numbers fail, including other numbers besides prime numbers. The cKdaKIos, or sieve, with some
other fragments of the writings of Eratosthenes, was printed with the "Phsenomena
of Aratus" at Oxford in 1672.
Poetius, at the end of his Arithmetic, published at Leipsic in 1728, printed a
table of the prime numbers and the factors of the composite numbers from 1 to
10,000. A similar table was published at Leyden by H. M. Anjema in 1767.
A larger table, extending from 1 to 101,000, was published at Halle by M. Kruger,
at the end of his "Thoughts on Algebra." Lambert extended this table as far as
102,000, and reprinted it, with other tables, at Berlin, in 1770.
Peter Barlow, of the Royal Military Academy, published in 1814 a volume of
Mathematical Tables. Table I. contains the prime numbers, and the factors of composite numbers from 1 to 10,000; also the reciprocals of these numbers calculated to
ten places of decimals. It contains also the squares and cubes of these numbers
with their square roots and cube roots each to seven places of decimals. Table V.
is a register of all the prime numbers between 1 and 100,109.