4 WOODALL, Merseniies Nuvihers. 



The author has been engaged for some time, in 

 conjunction with Lt.-Col. Cuimingham, R.E., in deter- 

 mining for all primes up to 100,000, the least exponents 

 (^,) which substituted in the expression 2?— i, give 

 numerical results exactly divisible by those prime numbers. 

 When he came to the prime 43441, he found that the 

 only possible ^'s were 181, and 5 x 181. 



The work then proceeded thus : — 



2^' divided by 43441 gave a remainder + 2480 : 

 squaring this, he found that 



2"" similarly gave a remainder + 252 19 : 

 again, squaring and doubling, 



2'''' gave a remainder of + r, 



which proved that 2'*'— i is exactly divisible by the prime 

 43441. On performing the division (directly) the quotient 

 was found to be 



70555 26073 82835 96526 70735 14491 06106379629341966511. 



About this quotient the only property that can be 

 asserted is that " if it has a factor it must be of one of 

 the two forms 



1448//+1 or 1448^+ 1087." 



As, for the 15 supposed composite (but still unverified) 

 numbers, all possible factors up to the comparatively high 

 limit of 500,000 have now been examined, it is obvious 

 that further work will be slow. 



The second column of Table I. shows all the prime 

 factors of Mersenne's numbers as now known. 



The Table II. gives the forms of the only possible 

 divisors of all numbers not yet completely examined. 



