Using the known results on Mersenne's numbers, we shall see that the first nine even perfect numbers are obtained by putting

in

and are

and

6, 28, 496, 8128, 33550336, 8589869056,

137438691328, 2305843008139952128,

2658455989570131744644692615953846176.

The method used by Cole in sifting prime numbers may be illustrated by considering the form of the factors, if any, of 421. This number may be expressed by quadratic partitions as

The discriminant of these quadratic expressions, 3, 5, 7, 11, 17, are all quadratic residues of 421.

Consider the quadratic residue 3.

I for p a factor of 421.

Take

Combine

This gives

= + 1 if p = 4 n + 1,

= - - 1 if p = 4 n + 3.

p=3n+1.

p=3n+1 with p = 4n+1, p = 4 n + 3.

p = 12n+1, and p = 12 n + II.

Now consider the quadratic residue 7 to further limit the forms of the factors.

The quadratic residues of 7 are 1, 2, 4, and the quadratic nonresidues are 3, 5, 6,

Combining the last forms with p = 4 n + 1, 3 gives

Combining with p = 60n+1, 11, 49, gives

p = 420 n + 1, 109, 121, 131, 169, 251, 289, 311, 361.

Р

Since no factors larger than the square root of the number investigated need be tried, the only form for a factor of 421 is p = 420 n + 1, where n = 0, I.

Therefore 421 is prime.

THE RECIPROCAL OF FERMAT'S THEOREM

The reciprocal of Fermat's theorem states that if a2 = I mod m

I,

In testing to see if a* = 1 mod m, where x is a factor of m it is not necessary to take for x each of the (k1 + 1) . . . (k, + 1) factors.

...

In fact, the Reciprocal of Fermat's Theorem may be modified to read as follows:

m I is the product of all the factors of m I but a. There

α;

fore if

m-1

a a I mod m, then

a* I mod m, where x is any combination of any or all the factors

then by raising both sides of this congruence to a suitable power, we obtain

Let us see how this theorem applies to the investigation of 2891, a Mersenne number.

and, therefore, 3 is a quadratic non-residue of 289 - I, also of

Hence the above theorem may be modified for the case of 289 — I as follows:

for

288

α;

a;= 3, 5, 17, 23, 89, 353, 397, 683, 2113, 2931542417,

we do not know if 289 I is prime or composite; if

The following is probably the most convenient method of carrying out the calculation for 289 - 1:

This method would, in general, involve about the same amount of computation as the method of Lucas."

• See p. 7.

A Quantitative Expression of the Periodic Classification of the Elements*

IN

FREDERICK GRAY JACKSON,

Instructor in Chemistry, University of North Dakota

N the periodic classification of the elements, as at present published in the textbooks, the elements are placed in squares or directly beneath each other. The vertical and horizontal periodicity is thus clearly brought out, but no consideration is given to the numerical difference between the atomic weight of any one element and that of its neighbors.

It became interesting to investigate these numerical differences in view of the large number of very accurate determinations of atomic weights that have recently been made. Speculations as to the reasons for these differences have been rife for many years, and many ingenious and elaborate tables have been published. Venable has written their history in his book on "The Development of the Periodic Law."

Some points not hitherto emphasized, can be shown if we plot the elements along a line, placing them in regard to a zero point in exact proportion to their atomic weights. In order to preserve the form of the periodic classification, let us stop after flourine, and begin a second line with neon. The same abscissa are preserved, and a constant whole number is subtracted from the atomic weights of the second row. We thus get a row of elements that do not come exactly beneath their homologues. Beginning with potassium, the third row is placed the same distance beneath the second, and plotted by subtracting a larger constant. This can be continued until the table is completed. Such a table was suggested eleven years ago by Prof. T. W. Richards, in the course of a series of lectures at Harvard University, but the subject was not carried out beyond an outline.

* Presented in abstract at the Minneapolis meeting of the American Chemica Society.

1 Chemical Publishing Company, 1906.

2 All atomic weights used in this paper are taken from the Report of the International Committee on Atomic Weights for 1911.