On a New Method of Factorization

VOL. 11, 192.5

MA THEMA TICS: D. N. LEHMER

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ON A NEW METHOD OF FACTORIZATION By DYIRRICK N. LIIHMI3R

DZPARTMBNT OF MATHZMATICS, UNIV1RSITY OP CALIFORNIA Communicated November 17, 1924

For numbers beyond the range of existing factor tables the problem of factorization is a very serious one, especially if nothing is known in advance concerning the character of the possible divisors. The most effective method so far invented is probably that used by Legendre (Th6orie des Nombres, 1797, p. 313) which is based on the theory of quadratic residues and makes use of the fact that all numbers having a given quadratic residue contain only such prime divisors as belong to certain linear forms, tables of which have been given by Legendre for various residues less in absolute value than 100. Thus if- lis known to be a residue the prime factors are all of the form 4n + 1, and such numbers as 3, 7, 11, 19, etc., are ruled out as possible divisors; or if 7 is known to be a residue the prime divisors must be of the form 28n + 1, 3, 9, 19, 25, 27. If -1 and 7 are both known to be residues these linear forms may be combined into 28n + 1, 9,25. Unfortunately the number of forms increases with the residue so that combining them is very laborious except for the smaller residues of the table. The larger residues are just as effective in excluding trial divisors, but they are often more trouble to use than they are worth because of the number of forms involved. These serious difficulties may be avoided by replacing the tables 6f linear forms by stencils. The combination of any number of sets of forms is then accomplished by piling the corresponding stencils, one on top of the other. A large residue will be as easily used as a small one, and the list of possible prime divisors is given automatically and with very little chance of error. It is proposed to extend Legendre's table to include all numbers less in absolute value than iOOO which contain no square factors other than unity. The difficulty of finding suitable residues will thus be very much diminished. This will involve the making of some 1216 stencils, but the labor of making a stencil will not be great, and when complete the device will furnish a practicable means of handling any number lessthan 2,361,279,649, and will be of great service in dealing with numbers far beyond this limit. Briefly the stencil method is as follows: Each stencil will have 100 rows and 50 columns, thus furnishing a cell for each of the 5000 primes from 1 to 48,593 listed on the first page of my List of Primes (Carnegie Institution of Washington, 1914). For any residue R a stencil is made by punching out a hole in each cell which corresponds to a prime which is of the proper linear form to have R for a quadratic residue. Thus for R = -1 a stencil will be made with holes punched in those cells which correspond to primes

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MA THEMA TICS: A. D. MICHAL

PROC. N. A. S.

which are of the form 4n + 1 and for R = 7 those cells are punched out which correspond to primes of the forms 28n + 1, 3, 9, 19,25,27. When these stencils are superposed the holes which shine through both stencils correspond to primes which belong to both sets of forms. The first stencil will shut out approximately one half the list of primes, and the second will shut out approximately half of those that remain; so that the discovery of as many as 12 residues will serve to exclude all but about 12 trial divisors for a number as great as the square of 48,593 or 2,361,279,649. For numbers of this order a sufficient number of quadratic residues less in absolute value than 1000 will not be difficult to obtain by well-known methods, the most effective of which is by expanding the square root of the number in a continued fraction. A machine for cutting the stencils from the master-stencil is easily devised. The labor of proof-reading or of type-setting with all the errors involved in those operations is completely eliminated. The device has one very important advantage over a factor table in that while a factor table is of no service at all for a number just beyond its range the system of stencils will be very useful for numbers far beyond the limit indicated. Moreover, any additional information regarding the character of the factors is not difficult to make use of, which advantage is not enjoyed by other powerful methods such as the representation of the number by binary quadratic forms.

FUNCTIONALS OF CURVES ADMITTING ONE-PARAMETER GROUPS OF INFINITESIMAL POINT TRANSFORMATIONS' By ARIsToTLo D. MIcHAi, DUPARTM1ENT Olt MATHEMATICS, THE Rics INs5ITiuT Communicated November 5, 1924

The purpose of this abstract is to study functionals of closed plane curves admitting one-parameter continuous groups of infinitesimal point transformations. Necessary and sufficient conditions are given. Extensions and generalizations of this invariant theory to functionals of closed plane curves depending in particular on a point on the curve and to functionals of n - 1 spreads in n dimensions are indicated. The theory of functional invariants may be divided into two parts: one, which seeks functionals admitting one-parameter groups of functional infinitesimal transformations ;2 and another, the object of this paper, which seeks functionals admitting one-parameter groups of ordinary infinitesimal transformations.3