Dec. 15, 1887] 



NATURE 



^':iZ 



being a less hackneyed letter than Euler's <^, which has no 

 claim to preference over any other letter of the Greek 

 alphabet, but rather the reverse. 



It is easy to prove that the half of any perfect number 

 must exceed in magnitude its totient. 



Hence, since i . i is less than 2, it follows that no 

 2 4 



odd two-fold perfect number exists. 



Similarly, the fact of 3 . 7 y being iggg than 2 is 

 2 6 10 

 sufficient to show that 3, 5 must be the two least elements 



of any three-fold perfect number; furthermore, -^ - . — 



2 4 16 



being less than 2, shows that 11 or 13 must be the third 



element of any such number if it exists ^ — each of which 



hypotheses admits of an easy disproof. But to disprove 



the existence of a four-fold perfect number by my actual 



method makes a somewhat long and intricate, but still 



highly interesting, investigation of a multitude of special 



cases. I ho\>t,iiu»nne/avente, sooner or later to discover 



a general principle which may serve as a key to a universal 



proof of the non-existence of any other than the Euclidean 



perfect numbers, for a prolonged meditation on the subject 



has satisfied me that the existence of any one such — its 



escape, so to say, from the complex web of conditions 



which hem it in on all sides — would be little short of a 



miracle. Thus then there seems every reason to believe 



that Euclid's perfect numbers are the only perfect 



numbers which exist ! 



In the higher theory of congruences (see Serret's 

 **Cours d'Algebre Supdrieure") there is frequent occasion 

 to speak of "a number n which does not contain any 

 prime factor other than those which are contained in 

 another number M." 



In the new nomenclature n would be defined as a 

 number whose elements are all of them elements o/M. 



As tN is used to denote the totient of N, so we may 

 use /xN to denote its multiplicity, and then a well-known 

 theorem in congruences may be expressed as follows. 



7Vie number of solutions of the congruence 



x- - I = o (mod P) 



is 2"^ if P is odd, 



2'^ -I if p is the double of an odd number, 

 2'' if P is the quadruple of an odd number, 



and 2"^"*'' in every other case. 



In the memoir above referred to, Euler says that no 

 one has demonstrated whether or not any odd perfect 

 numbers exist. I have found a method for determining 

 what (if any) odd perfect numbers exist of any specified 

 order of manifoldness. Thus, e._^., I have proved that 

 there exist no perfect odd numbers of the ist, 2nd, 3rd, 

 or 4th orders of manifoldness, or in other words, no odd 

 primary, binary, ternary, or quaternary number can be a 

 perfect number. Had any such existed, my method must 

 infallibly have dragged each of them to light 2 



In connection with the theory of perfect numbers I 

 have found it useful to denote^' — i when p and / are 

 left general as the Fermatian function, and when p and i 

 have specific values as the nh Fermatian of/. In such 

 case/ may be called the base, and / the index of the 

 Fermatian. 



' 3< 5. 7 can never co-exist as elements in any perfect nuoiberas shown by 

 the fact that 'Jl'jti! . JL±_5_ . LiL? + 49 ; ,-.<,. l6/ ^ £ ^ ^\ .^ 



9 5 49 15 7 40' 



greater than 2. Thus we see that no perfect number can be a multiple 



of 105. So again the fact that 5 . 7 _ £i _ 13 _ U _ £9 j^ i^^^ jjjan 2 is suffi- 



4 6 10 12 i6 18 

 cicm to prove that any odd perfect number of multiplicity less than 7 must 

 be divisible by 3. 



* I have, since the ab )ve was in print, extended the proof to qu'nary 

 numbers, and anticipate no d.fRculty in doing so for numbers of higher 

 degrees of .multiplicity, so that it is to be hoped that the way is now paved 

 towards obtaining a general proof of this/rt/wary theorem. 



Then we may express Fermat's theorem by saying that 

 either the Fermatian itself whose index is one unit below 

 a given prime or else its base must be divisible by that 

 prime} 



It is also convenient to speak of a Fermatian divided 

 by the excess of its base above unity as a Reduced Ferma- 

 tian and of that excess itself as the Reducing Factor. 



The spirit of my actual method of disproving the exist- 

 ence of odd perfect numbers consists in showing that an 

 «-fold perfect number must have more than « elements, 

 which is absurd. The chief instruments of the investigation 

 are the two inequalities to which the elements of any per- 

 fect number must be subject and the properties of the 

 prime divisors of a Reduced F"ermatian with an odd 

 prime index. 



New College, November 28. J. J. Sylvester. 



COUTTS TROTTER. 



A GREAT calamity has fallen on the University of 

 -^^ Cambridge and on Trinity College, and many men 

 differing widely in their interests and callings are bearing 

 together the burden of a common sorrow in the knowledge 

 that the Rev. Coutts Trotter, the Vice-Master of Trinity 

 College, was no more. Mr. Trotter suffered from a severe 

 and prolonged illness during last winter and early spring, 

 and though in the summer he seemed to have almost 

 regained his health, he began as the year advanced 

 once more to lose ground. When he returned from 

 abroad in October his condition gave rise to great 

 anxiety among his friends ; as the term went on he grew 

 worse rather than better ; and an attack of inflammation 

 of the lungs rapidly brought about the end, which took 

 place in his rooms in College, in the early morning of 

 Sunday, December 4. 



During the last twenty of the fifty years of Mr. Trotter's 

 life both the University of Cambridge and Trinity College 

 have undergone great and important changes. In bring- 

 ing about these changes Mr. Trotter had a great share, 

 perhaps a greater share than any other individual 

 member of the University ; and while those changes are 

 probably neither wholly good nor wholly evil, but good 

 mixed with evil, no one hand, as the changes were being 

 wrought, did so much good and so little evil as his. A. 

 wide and yet accurate knowledge of many different 

 branches of learning, a genuine sympathy with both 

 science and scholarship, a judicial habit of mind which 

 enabled him to keep in view at the same time broad issues 

 and intricate details, a clear insight into the strength and 

 weakness of academic organization, and a singular skill 

 in drafting formal regulations, — these qualities, aided by a 

 kindly courtesy which disarmed opponents, and a patience 

 which nothing except perhaps coarse rudeness could ruffle, 

 enabled him in his all too short life to do for his College 

 and for his University more than it seemed possible for 

 one man to do. 



The academic labours which thus year by year increased 

 upon him, though they in many ways, both directly and 

 indirectly, tended to the advancement of science, became, 

 increasingly, hindrances to his pursuing actively any 

 special path of scientific inquiry, as he had once hoped 

 to do. His love of science began with his boyhood, while 

 he listened to the Royal Institution lectures of Faraday. 

 Having taken a degree, with honours in both classics 

 jmd mathematics, and having obtained a Fellowship at 

 Trinity, he gave up to scientific study much of the leisure 

 thus afforded to him : and, in order more thoroughly to train 

 himself, spent the best part of two years at Heidelberg, 

 during a portion of which time he was engaged in physio- 

 logical research under Helmholtz. He acquired a very 



' So too we may state the important theore-n that if an element of a 

 Fermatian is its index the component which has that index for its base 

 must be its square. 



