Manchester Memoirs, Vol. Ivi. . 191 1 ), No. f. 



I. Mersenne's Numbers. 



By H. j. WOODALL, A.R.C.Sc. (I.on.). 



[A'ccdizwd Ottol er Olh, igu . Mead October ijth, i<piJ.) 



[The author desires to express his thanks to I.t.-Col. 

 Cunningham, R.E., for help in writing this note.] 



In 1644 Mersenne made the assertion that the only 

 values of />, not greater than 257, which make 2^—1 a 

 prime are : — 



1. 2, 3, 5, 7, 13, 17, 19, ^i, 67, 127, 257. 



He published no proof of his statement, nor any 

 indication as to how he arrived at it. Up to the present 

 time it has only been partly verified. Only two errors 

 have been found, viz. : — 



(2"' - i) found prime ; 

 (2"^- i) found composite. 



The work so far done leaves still unverified only 15 out of 

 the 44 numbers originall)' affirmed composite (by Mer- 

 senne), and one affirmed prime, viz. : — 



composite /=ior, 103, 107, 109, 137, 139, 149, 157, 167, 



'73' '93. '99> 227, 229, 241 ; 

 and prime /'--^257. 



Table I. shows the present state of the examination 

 of this assertion. 



The purpose of this paper is to put on record the 

 method by which 2'^'— i was proved to be composite. 



December 12th. igii. 



