150 JOURNAL OF THE WASHINGTON ACADEMY OF SCIENCES VOL. 13, No. 8 
MATHEMATICS.—A_ remarkable formula for prime numbers. 
Patt R. Hey, Bureau of Standards. 
n—1 
The expression —is known to be integral for all odd prime 
values of n and non-integral for all even values of n. Numerical test 
shows that it is also non-integral for all odd composites up to n =1000 
with the exception of 341 and 645. In other words, using this ex- 
pression as a test for the prime or composite nature of a number, its 
indication that the number is composite is sound; but its indication 
that a number is prime is uncertain. 
In order to handle the very large numbers to which such a use of 
this formula gives rise, we make use of the principle that if the product 
of several numbers A BC..... be divisible by n with a certain 
remainder, the same remainder will be obtained if for any or all of 
the quantities A, B, C .... we substitute a, b,c..... their 
separate remainders when singly divided by n. For example: 
17 x2 
3 
gives remainder 1 as does also ax x 2 being the remainder 
of 17 divided by 3. 
In general if 
A=a-+nx 
B=b+ny 
C=C+nz ete, 
then AB Cn aes Le ee i + terms containing n, and the 
remainder of the whole expression after division by n will be the re- 
mainder resulting from the first term abc..... 
In the practical application of this formula one needs a calculating 
machine and a table of powers of 2. Such a table has been constructed 
by the author up to 21, and blue print copies of it will be furnished 
to any one to whom it would be useful. 
As an example: 
n = 483 =3 X7 X23 
482 
Remainder of - a should equal 0 if 483 is prime, which is 
482 
ival ind = 
equivalent to remainder 483 1 
94st — (224) 20 x 22 
1 Received January 14, 1923, Presented at the Annual Meeting of the Philosophical 
Society. 
