614 



REPORT — 1 895. 



8. On Mersenne's Numbers. 

 By Lt.-Col. Allan Cunningham, R.E., Hon. Fell. King's Coll. Loud. 



A Mersenne's number is one of form N = (2'' — 1), where y is a prime. Divisors 

 of these are difficult to discover. Their prime divisor (;;), when N is composite, 

 must be of form p = 2egi + 1, and also of one of forms p = 8i±l, and 2 must be a 

 residue of order e. 



Simple rules (due to Legendre, Gauss/ and Jacobi,) are given for finding directly 

 divisors (p) — when such exist — for the cases of 2e = 2, 6, 8, 16, 24. An indirect 

 method (due to JMr. C. E. Bickmore) is also given for the case when p = 2ee'. g + 1, 

 where 2e has any of the above values, and e' = an odd number >3. 



A table of divisors (p) — the greater part of which is believed to be original — 

 is given : this is believed to be complete for all primes ^;< 15,000 for the simple 

 cases of 2e = 2, 6, 8, 16, 24. The followiug thirteen new instances were discovered 

 by the indirect method quoted, and are the outcome of much labour. 



One of these, viz. (2^^^ — 1), is among those stated by Mersenne in 1,664, but with- 

 out proof, to be composite ; proof of this is now supplied in the discovery of a 

 divisor (p = 7,487). 



Nineteen of the Mersenne's numbers stated by Mersenne to be composite (viz., 

 when 9 = 71, 89, 101, 103, 107, 109, 137, 139, 149, 157, 163, 167, 173, 181, 193, 

 199, 227, 229, 241), and three of those stated by him to be prime (viz., when q = 67, 

 127, 257), remain st.ill xmverified. The author has tried all prime numbers < 50,000 

 without finding a divisor for any of them. 



There are only \eu prime Mersenne's numbers known, viz., when q= 1, 2, 3, 5, 

 7. 13, 17, 19, 31, 61 ; the establishment of any more is very difficult. It is worth 

 noting that these ten values of q, as also three more {q = 07, 127, 257) conjectured by 

 Mersenne to yield prime values of N, all fall under one of the /oew- forms q = 2'^ ± 1, 

 or 2^ + 3 ; but it is not true that such values of q necessarily make N prime. 



9. Recent Bevelojyments of the Lunar Theory. By P. H. Cowell, M.A. 



Kepler discovered that the motion of planets about the sun and the moon 

 about the earth took place approximately in ellipses, and Newton showed that 

 motion in an ellip.se according to Kepler's well-known laws was the consequence 

 of the law of gravitation. 



For nearly two centuries after Newton, the lunar theorists based their investi- 

 gations of the moon's motion upon Newton's discovery. Their reason for doing this 

 was that the elliptic inequality is by far the largest of all the lunar inequalities. 

 The other inequalities were then calculated as disturbances due to the sun. One 

 modification had, however, to be made. In order to agree with observations, 

 displacements increasing with the time had to be assigned to the apse and node. 

 The orbit thus modified no longer satisfied the undisturbed equations of motion, 

 and the velocities of the node and apse, as well as the remaining inequalities, had 

 to be found — in most theories — by continued approximation. 



In performing the algebraical calculations, however, it appeared that the terms 

 constituting the inequality known as the variation had first to be calculated, and 



' The author is indebted to Mr. C. E. Bickmore for the communication of these 

 rules. 



