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McCoy, Elizabeth (ed.) / Transactions of the Wisconsin Academy of Sciences, Arts and Letters
volume LXII (1974)

Marshall, Arthur
Primes and Farey sequences,   pp. 245-246 PDF (607.9 KB)

Page 245

```PRIMES AND FAREY SEQUENCES 245
Arthur Marshall
This paper is aimed at helping teachers of pre-college or freshman mathematics
to demonstrate how prime numbers arise in the natural ("counting")
number
system and how prime numbers can be "predicted" from examination
of numbers
less than themselves.
We will show that primes are the result of adding the numerator and denominator
of certain "Farey Sequences" fractions (a/b wi'th 0 &lt; a
&lt; b, b  2,
and (a, b) = 1), starting with 1/2. (We have just used the greatest-common-divisor
symbol of number theory literature.)
All letters stand for positive integers (sometimes called natural numbers
or counting numbers).
THEOREM: If (a, b) = 1, then (a, a + b) = 1 and
(b, a + b) = 1. Conversely, if 0 &lt; a &lt; n/2, and (a, n) = 1,
then
(a, n &mdash; a) = 1 with a/ (n &mdash; a) &lt; 1, and (n &mdash;
a, n) =
1, with
(n &mdash; a)/n &lt; 1.
PROOF: If a = 1, ' the theorem is obvious.
If a &gt; 1, then (using congruence notation of elementary number theory)
a = 0 (mod p), where p is any prime dividing a. Because (a, b) = 1, b ~4
0 (mod p), then, by addition of congruences (a + b) ~ 0 (mod p), and the
first assertion of the first part of the theorem is proved. We follow an
analogous procedure for q, any prime dividing b, and the second assertion
of the first part is proved.
To prove the converse, we note that p, any prime dividing a, cannot, by
hypothesis divide n. Then (n &mdash; a) = n ~ 0 (mod p) and the first
statement
of the converse is proved. To prove the second statement of the converse,
we no'te that q, any prime dividing ii, cannot, by hypothesis divide (n &mdash;
a), because (n &mdash; a) =
&mdash;a ~ (mod q), and that completes proof of the theorem.
COROLLARY 1. All fractions in Farey sequences, lying between 0 and 1, can
be obtained by successive additions of numerators and denominators, beginning
with &frac12;~ to obtain new fraction denominators and taking each summand
numerator and denominator as numerators over the ne,w denominators. (The
proof is immediate from the theorem.)

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