552 



REPORT — 1901. 



7. On Idoneal Numhers. 

 By Lt.-Col. Allan Cunningham, R.E., and the Rev. J. Cullen, S.J. 



About the year 1778 Euler discovered tlie existence of a class of positive 

 numbers {inn), such that if an odd number N be expressible in only one way in the 

 form (wii- + ny"^) [with mx prime to ny'\ or (,r- - mny-) [with .r prime to mny\ 

 then N is either a prime or the square of a prime. These numbers he styled numeri 

 idonei from their special fitness to aid in the detection of higrh primes. He gave 

 rules for their discovery, and actually discovered sixty-five of them, the largest 

 being 1848, and stated that there are no more > 1,848 but < 10,000. The joint 

 authors have recently extended their search up to 101,220 by a sort of graphic 

 process of solving simultaneous linear congruences (invented by the Kev. J. Cullen), 

 with the result that no more such numbers exist > 1,848, but < 101,220. (The 

 whole of the work ending in this result has been done independently by each of 

 the joint authors.) 



As to the forms {mx- 'v nj/-), {x- '^ mny-) mere automorphs of the same form, 

 i.e., products of the form by its unit-form t~ — mnv' = 1, are not to he considered 

 as distinct forms. With this proviso it is found that negative idoneale ( - mti) 

 are very numerous. Gauss's tables show that, excluding squares, all but thirty- 

 five of the numbers < 328 yield negative idoneals. Also the authors find that all 

 the known positive idoneals (except 37) are also negative idoneals. 



Several new theorems on quadratic forms whose determinant is an idoneal 

 were announced. 



As an application all the odd numbers of form (i'^ + 1848?/''), wherein x is 

 prime to 1 848 y, from 10,000,000 to 10,100,000. have been examined; the 189 

 numbers shown below were found to be expressible in only one way in that form, 

 and (squares having been excluded) are therefore all Primes. 



[This work was done by two computers independently under Colonel Cunning- 

 ham's supervision.] 



