or "cross-talk." As it is only the small unbalanced capacity that is required to be known, it is mesured by a similar but simpler bridge method than that described above. The balancing capacity is a 500-mmf. revolving air condenser with a large scale that may be read to micro-microfarads. No new principles are involved so this bridge will not be described in detail. The tests described above are especially designed for cables, but it is obvious that they may be applied to general laboratory work, and for the mesurement of very small capacities the bridge method with careful shielding becomes a method of the highest precision. Mersenne's Numbers and the Reciprocal of Fermat's Theorem RAYMOND ROYCE HITCHCOCK, Instructor in Mathematics, University of North Dakota ROFESSOR DOUTHAT of the University of West Virginia says in his treatise, "A New Method of Obtaining all Possible Prime Numbers from One to Infinity," "God gave man the digits, and taught him how to use them, in order that thru them man might be led to use these forms; first, to perfect all commercial transactions; second, in solving the chemical, physical and electrical problems of earth; third, in the study of immensity and grandeur as revealed in the heavens. Such ideas of continuance can be attained only thru calculations in space. "The one subject connected with arithmetic, on which all arithmeticians so far have failed, is analysis. They seem to have partaken in common with ordinary people of the erroneous idea that numbers, like sticks and stones, belong just where you put them and that the only way to find out how to deal with any large number is to try experiments. This latter operation of experimentation is proof of the want of proper analysis." It was the mathematician, Mersenne, who, in 1644, asserted that out of the 56 primes not greater than 257, there were only 12 primes which, taken as exponent (p), make the number N = 2o I, also prime. No proof was published, and even up to now, this statement has been only partially verified. The resolution into numerical factors of an I, when a and n denote positive integers, is a problem to which students of higher arithmetic, especially in past time, have devoted much attention. The factors of 2o I are of especial interest for two reasons: first, because of Mersenne's assertion as to 2" - 1 being a prime for certain values of p less than 257; and, secondly, because of their bearing on the product 2o which is not always equal to 1, but for certain values of p is the product of a number of similar products, each equal to 1. The best known example of this is which is proved at once by putting the first product equal to X and noting that X. 2 sin If p31, or p = 43, the original product resolves into three П others, as is readily seen by beginning with or as argument of the cosine and successively doubling; then since 31 43 25 = 1 (mod. 31) and one of the three products 27 = 1 (mod. 43), for 31 and 43 is obtained. Mersenne1 asserted, in 1644, that the only values of p not greater than 257 that make 2o I a prime are 1, 2, 3, 5, 7, 13, 17, 19,31, 67, 127 and 257, to which list Seelhoff has shown that we must add 61, and Cole3 has shown that we must strike out 67. The method by which Mersenne determined the values of p is lost, and its discovery remains among the riddles of higher arithmetic. Lucas has estimated that in order to verify the last assertion of Mersenne's, that is, that 2257 1 is a prime, by known methods other than his, the whole population of the globe 1 Mess. of Math. (W. W. R. Ball), Vol. XXI, pp. 34–40. • Ibid. Am. Math. Soc., Vol. X, p. 134. calculating simultaneously would require more than a million of millions of millions of centuries." Mersenne's statement, regarding the value of p, has been verified for all excepting 21 values of p, namely, 89, 101, 103, 107, 109, 127, 137, 139, 149, 157, 163, 167, 173, 181, 193, 197, 227, 229, 241 and 257. = It remains to prove that p = 127 and p 257 make 2P I prime, and that the other values here given make 2o - I composite. If p is not a prime then it is evident that 2" I is composite, and two or more of its factors can be written by inspection. The factors of such values of 2" I as are less than a million can be verified easily. The table on the following page gives the investigations to date. In his letter, Fermat had said that the only possible prime factors of 2o± 1, when p was a prime, were of the form np + 1, where n is an integer or zero. In 1748, Euler proved this statement and added that since 2o± 1 is odd, every factor of it must be odd and therefore if p is odd n must be even. The proof follows: Euler showed that if T-I = 2 np, π = 2 np + 1. 1 231 I had any prime factors they had to be of the form 248 n + 1 or 248 n + 63, and had to be less than √231 I, that is than 46339. It was necessary to try only 40 divisors to see if 231 I is prime or composite. Plana used a similar method in his study of 241 I. The prime factors of this number, if any, are of the form 328 n + I or 328 n + 247, and lie between 1231 and √241 — I, that is, 1048573. There are 513 such divisors, and the seventeenth of these gave the required factors. This method is too laborious to use for values of p greater than 41. Lucas proved a proposition to the effect that if 4 n + 3 and 8 n + 7 are primes, then 24n+3 I o mod 8 n + 7. Am. Jour. of Math., Vol. VI, p. 236. |