R e e R P e
tithialy e TNt

 

T TR i S i

é,
E
§
S
-
ks
S

58 _ THE FASCINATION OF NUMBERS

whereby specific numbers may be tested for primality. It
may indeed reach further, pointing the way to the yet un-
discovered general solution.

If we approach the subject from a different angle and
assume that a prime might possibly be the sum of a series,
we can at once dimiss the possibility where the number of
terms in the series is even, since the sum of an even number of
terms must be even and is therefore not a prime. It is, in
fact, impossible for a prime to be the sum of a series for if it
were it would be of the form Eah and would have the

2
a+L
factors # and (——{2;—2
We are, however, assuming that a prime might be the sum
of a series of odd numbers. If we select an odd number, N,
then assuming that this can be represented as the sum of a
series then:

Nzn(a +L)
2

But L=[a+(n—1)d]
S0 N=—Z—[2a+(n—1)d]
and since d=2
being the difference between consecutive odd numbers,
then N=n(a+n—1)

=n%+(a—1)n

whence #72+(a—1)n—N=o0
We now have the supposed relationship expressed as a
quadratic equation. If it has any rational solution at all, this
can be found by the general solution formula for quadratic
equations, whence:
W —(a—1) :!-_-\/(a——l)?r—’(-z}.fif
2
—a+14+Vat—2a44N+1
2