March i, 1888J 



NATURE 



417 



Report of the British Association, in which (p. 668) further 

 experiments by Dr. A. Scott are reported. Dr. Scott has 

 succeeded in reducing the amount of nitrogen present as impurity 

 to I part in 15,000, and the ratio of hydrogen to oxygen which he 

 calculates from the newer and more accurate experiments is 

 1*996 or 1*997 to I "000. This ratio agrees very well with that 

 deduced by me from the older experiments, but is considerably 

 higher than the ratio previously adopted by Dr. Scott, and 

 quoted by Prof. Thorpe in his article on the comp.jsition of 

 water. Sydney Young. 



Univeriity College, Bristol. 



ON TRE DIVISORS OF THE SUM OF A GEO- 

 METRICAL SERIES WHOSE FIRST TERM 

 IS UNITY AND COMMON RA TIO ANY POSI- 

 TIVE OR NEGA TIVE INTEGER. 



"Nein! 



Wir sind Dichter." 1 



— Kronecker in Berlin. 



A 



REDUCED Fermatian.^ , is obviouslv only 



' r-i - ^ 



another name for the sum of a geometrical series 



whose first term is unity and common ratio an integer, r. 



\i p is a prime number, it is easily seen that the above 

 reduced Fermatian will not be divisible by j^J, unless r~\ 

 is so, in which case (unless / is 2) it will bs divisible by 

 p, but not by />-. 



This is the theorem which I meant to express in the 

 footnote to the second column of this journal for December 

 15, 1887, p. 153, but by an oversight, committed in the act 

 of committing the idea to paper, the expression there 

 given to it is erroneous. 



Following up this simple and almost self-evident 

 theorem, I have been led to a theory of the divisors of a 

 reduced Fermatian, and consequently of the Fermatian 

 itself, which very far transcends in completeness the 

 condition in which the subject was left by Euler (see 

 Legendre's " Theory of Numbers," 3rd edition, vol. i., 

 chap. 2, § 5, pp. 223-27, of Maser's literal translation, 

 Leipzig, 1886),^ and must, I think, in many particulars be 

 here stated for the first time. This theory was called for 

 to overcome certain difficulties which beset my phantom- 

 chase in the chimerical region haunted by those doubtful 

 or supposititious entities called odd perfect numbers. Who- 

 ever shall succeed in demonstrating their absolute non- 

 existence will have solved a problem of the ages comparable 

 in difficulty to that which previously to the labours of 

 Hermite and Lindemann (whom I am wont to call the 

 Vanquisher of PI, a prouder title in my eyes than if he 

 had been the conqueror at Solferino or Sadowa) environed 

 the subject of the quadrature of the circle. Lambert had 

 proved that the Ludolphian * number could not be a 



• Such were the pregnant words recently uttered by the youngest of 

 the splendid tr.umvirate of Berlin, when challsnged to declare if he still 

 held the opinion advanced in his early inaugural th;sis (to the effect 

 that riiathematic consists exclus^•ely in tie setting out of self-evident 

 truths, — in fact, amounts to no more than showing that two and two 

 make four), and maintained unflinchingly by him in the face of the elegant 

 raillery of the late M. Duhamel at a dinner in Paris, wh;re his inerro^jat. r 

 — the writer of these lines — was present. This doctoral thesis ought tj be 

 capable of being found in the archives of the University (I believe) of 

 Ureslau. 



^ The word Fermatian. formed in analogy with the words Hessian, 

 Jacobian, Pfaffian, Bezoutiant, Cayleyan, is derived from the name of 

 Fermat, to whom it owes its exisience among recognized algebraical forms. 



3 I find, not without surprise, that some of the theorems here produced, 

 including the one contained in th; corrected footnote, have been previously 

 stated by myself i.i a portion of a paper " On certain Ternary Cubic Form 

 Equati )ns,"en-itled "Excursus A — On the Divisors of Cyclotomic Functions" 

 {American Journal 0/ Mathematics, v A. ii., 1879, p. 357) the contents a .d 

 almost the existence of which I had forgotten : but the mode of presentation of 

 the theory is different, and I think clearer and more compact here than in th; 

 preceding paper ; the concluding theorem (which is the important one for 

 the theorj^ of perfect numbers) and the propositi )ns immediately leading up 

 to it in this, are undoubtedly not contained in the previous paper. 



I need hardly add that the term cyclotomic function is einployed to desig- 

 nate the core or primitive factor of a Fermatian, because the resolution into 

 factors of such function, whose index is a given number, is virtually the 

 same problem as to divide a circle into that number of equal parts. 



* So the Germans wisely name ir, after Ludolph van Ceulen, best known to 

 us by his second name, as the calculator of tt up to thirty-six places of 

 de:imals. 



fraction nor the square root of a fraction. Lindemann 

 within the last few years, standing on the shoulders of 

 Hermite, has succeeded in showing that it cannot be the 

 root of any algebraical equation with rational coefficients 

 (see Weierstrass' abridgment of Lindemann's method, 

 Sitzungsberichte der A.D. IV. Berlin, Dec. 3, 1885), 



It had already been shown by M. Servais (" Mathesis," 

 Lifege, October 1887), that no one-fold integer or two-fold 

 odd integer could be a perfect number, of which the proof 

 is extremely simple. The proof for three-fold and four- 

 fold numbers will be seen in articles of mine in the course 

 of publication in the Comples rendiis, and I have been 

 able also to extend the proof to five-fold numbers. I 

 have also proved that no odd number not divisible by 

 3 containing less than eight elements can be a perfect 

 number, and see my way to extending the proof to the 

 case of nine elements. 



How little had previously been done in this direction is 

 obvious from the fact that, in the paper by M. Servais 

 referred to, the non-existence of three-fold perfect numbers 

 is still considered as problematical ; for it contains a 

 " Theorem " that if such form of perfect number exists it 

 must be divisible by fifteen : the ascertained fact, as we 

 now know, being that this hypothetical theorem is the 

 first step in the reductio ad absurduni proof of the non- 

 existence of perfect numbers of this sort (see Nature, 

 December 15, 1887, p. 153, written before I knew of 

 M. Servais' paper, and recent numbers of the Comptes 

 rendiis). 



But after this digression it is time to return to the 

 subject of the numerical divisors of a reduced Fermatian. 



We know that it can be separated algebraically into as 

 many irreducible functions as there are divisors in the 

 index (unity not counting as a divisor, but a number 

 being counted as a divisor of itself), so that if the com- 

 ponents of the index be a", b^, c, . . . the number of such 

 functions augmented by unity is 



(a+l)(3+l)(X+l) 



All but one of these algebraical divisors, with the excep- 

 tion of a single one, will also be a divisor of some other 

 reduced Fermatian with a lower index : that one, the 

 core so to say (or, as it is more commonly called, the irre- 

 ducible primitive factor), I call a cyclotomic function of the 

 base, or, taken absolutely, a cyclotome whose index is the 

 index of the Fermatian in which it is contained. 



It is obvious that the whole infinite number of such 

 cyclotomes form a single infinite complex. Now it is of 

 high importance in the inquiry into the existability of 

 perfect numbers to ascertain under what circumstances 

 the divisors of the same reduced Fermatian, i.e. cyclotomes 

 of different indices to the same base can have any, and 

 what, numerical factor in common. For this purpose I 

 distinguish such divisors into superior or external and 

 inferior or internal divisors, the former being greater, and 

 the latter less, than the index. 



As regards the superior divisors, the rule is that any 

 one such cannot be other than a unilinear function of the 

 index (I call kx -j- i a unilinear function of x, and k the 

 unilinear coefficient) and that a prime number which is 

 a unilinear function of the index will be a divisor of the 

 cyclotome when the base in regard to the index as modu- 

 lus is congruous to a power of an integer whose exponent 

 is equal to the unilinear coefficient. 



As regards the inferior divisors, the case stands thus. 

 If the index is a prime, or the power of a prime, such 

 index will be itself a divisor. I f the index is not a prime, or 

 power of a prime, then the only possible internal divisor 

 is the largest element contained in the index, and such 

 element will not be a divisor unless it is a unilinear func- 

 tion of the product of the highest powers of all the other 

 elements contained in the index. 



It must be understood that such internal divisor in 



