July 12, 1888] 



NATURE 



261 



Taking the logarithms of both sides of the equation, we find 

 ©i - i© 2 + J®3 " • • • • > i lo g log N + i log M N - \ log 3, 

 where 0,- means the sum of *th powers of the reciprocals of 

 all the prime numbers, not exceeding N, of the form 6« + 5. 



Either from this equation or from the one from which it is 

 derived it at once follows that the number of primes of the form 

 6« + 5 is greater than any assignable limit. 



Parallel to what has been shown in the preceding case, if it 

 could be ascertained that the sum of the terms of the aggregate 

 Q . whose denominations do not exceed N bears a ratio which 

 becomes indefinitely small to the total aggregate, it would follow 

 by strict demonstration that the sum of the reciprocals of the 

 primes of the form 6« + 5 inferior to N would always differ by 

 a limited quantity from the half of the second logarithm of N. 



It is perhaps worthy of remark that the infinitude of primes 

 of the forms 4« + 3 and 6« + 5 may be regarded as a simple 

 rider to Euclid's proof (Book IX., Prop. 20) of the infinitude of 

 the number of primes in general. 



The point of this is somewhat blunted in the way in which it is 

 presented in our ordinary text-books on arithmetic and algebra. 



What Euclid gives is something more than this : 1 his statement 

 is, " There are more prime numbers than any proposed multitude 

 (■n\rj9os) of prime numbers" ; which he establishes by giving a 

 formula for finding at least one more than any proposed number. 

 He does not say, as our text-book writers do, "if possible let 

 A, B, . . . . (J be all the prime numbers," &c, but simply that 

 if A, B, .... C are any proposed prime numbers, one or 

 more additional ones may be found by adding unity to their 

 product which will either itself be a prime number, or contain 

 at least one additional prime ; which is all that can correctly be 

 said, inasmuch as the augmented product may be the power of a 

 prime. 



Thus from one prime number arbitrarily chosen, a progression 

 may be instituted in which one new prime number at least is 

 gained at each step, and so an indefinite number may be found 

 by Euclid's formula : e.g. 17 gives birth to 2 and 3 ; 2, 3, 17 to 

 103 ; 2, 3, 17, 103 to 7, 19, 79 ; and so on. 



We may vary Euclid's mode of generation and avoid the trans- 

 cendental process of decomposing a number into its prime factors 

 by using the more general formula, a, b, . . . . c + 1, where 

 a, b, . . . . c, are any numbers relatively prime to each other ; 

 for this formula will obviously be a prime number or contain one 

 or more distinct factors relatively prime to a, b, .... c. 



The effect of this process will be to generate a continued 

 series of numbers all of which remain prime to each other : if 

 we form the progression 



a, a + 1, a? + a + 1, a(a + i)(a 2 + a + 1) + 1, . . . . 



and call these successive numbers 



«,, Up « 3 , « 4 , . . . . 



we shall obviously have 



U x + j = M M 



- u x + I. 



It follows at once from Euclid's point of view that no primes 

 contained in any term up to u x can appear in u x + 1( so that all 

 the terms must be relatively prime to each other. The same 

 consequence follows a posteriori from the scale of relation above 

 given ; for, as I had occasion to observe in the Comptes rendus for 

 April 1888, if dealing only with rational integer polynomials, 



<p(x) = (x- a)f{x) + a, 



then, whatever value, c, we give to x, no two forms <t>*(c), <t> J (c) 

 can have any common measure not contained in a : in this case 

 <p(x) = (x - \)x + 1 ; so that <£'(<:) and <p'{c) must be relative 

 primes for all values of i andy'. 2 



It is worthy of remark that all the primes, other than 3, 

 implicitly obtained by this process will be of the form 6i + I. 



Euclid's own process, or the modified and less transcendental 

 one, may be applied in like manner to obtain a continual suc- 

 cession of primes of the form 4^+3 and 6« + 5. 



1 Whereas the English elementary book writers content themselves with 

 showing that to suppose the number of primes finite involves an absurdity, 

 Euclid shows how from any given prime or primes to generate an infinite 

 succession of primes. 



- Another theurem of a similar kind is that, whatever integer polynomial 

 <p(x) may be, if i, J have for their greatest common measure k, then 

 <(>*[<p(o)] will be the greatest common measure of <p l [<t>(o)], ^[^(o)]. 



As regards the former, we may use the formula 



2.a.b....c+i 



(where a, b, . . . . c are any "proposed " primes of the form 

 4« + 3), which will necessarily be of the form 4« + 3, and must 

 therefore contain one factor at least of that form. 

 As regards the latter, we may employ the formula 



3 . a . b . . . . c + 2 



(where a, b, . . . c are each of the form 6« + 5), which will 

 necessarily itrelf be, and therefore contain one factor at least, of 

 that form. 



The scale ot elation in the first of these cases will be, as 

 before, 



u x + ! = u x 2 - u x + 1 ; 



so that each term in the progression, abstracting 3, will be of 

 the form 4^ + 3 and 6/ + 1 conjointly, and consequently of the 

 form I2« + 7 ; as e.g., 



3, 7, 43, 1807, .... 

 In the latter case the scale of relation is 

 u x + x = zij - 2it x + 2, 



which is of the form (u x - 2)u x + 2. It is obvious that in each 

 progression at each step one new prime will be generated, and 

 thus the number of ascertained primes of the given form go on 

 indefinitely increasing, as also might be deduced a posteriori by 

 aid of the general formula above referred to from the scale of 

 relation applicable to each. Each term in the second case (the 

 term 3, if it appears, excepted) will be simultaneously of the 

 form 6i - I and 4/ + 1, and consequently of the form 12n + 5, 

 as in the example 5, 17, 257, 65537, .... 



The same simple considerations cease to apply to the genesis 

 of primes of the forms 4W + I, 6n + I. We may indeed apply 

 to them the formulae 



(2 . a . b . . . . cf + 1 and ${a . b . . . . c)~ + l 



respectively, but then we have to draw upon the theory of 

 quadratic forms in order to learn that their divisors are of the 

 form 4« + 1 and 6« + 1 respectively. 



Of course the difference in their favour is that in their case all 

 the divisors locked up in the successive terms of the two progres- 

 sions respectively are of the prescribed form ; whereas in the other 

 two progressions, whose theory admits of so much simpler treat- 

 ment, we can only be assured of the presence of one such factor 

 in each of the several terms. 



Euler has given the values of two infinite products, without 

 any evidence of their truth except such as according to the 

 lax method of dealing with series without regard to the laws of 

 convergence prevalent in his day, and still held in honour in 

 Cambridge down to the times of Peacock, De Morgan, and 

 Herschel inclusive (and this long after Abel had justly denounced 

 the use of divergent series as a crime against reason), was 

 erroneously supposed to amount to a proof, from which the same 

 consequences may be derived as shown in the foregoing pages, 

 and something more besides. * These two theorems are — 



W 3 + 1 ' 5 - 1 ' 7 + 1 ' 11 + 1 ' 13 - 1 ' 4 



(where, corresponding to the primes 3, 7> n> • • • • OI " the 

 form i,n + 3, the factors of the product on the left are 



3 7 II 



3 + 1' 7 + 1' 11 + 1 ' 

 all of them with the sign + in the denominator ; while the 

 fractions corresponding to primes of the form 4« + I have the 

 - sign in their denominators). 



_J 7_ 11 13 17 ( .=^V3 



5 + 1 " 7 - 1 ' 11 + 1 ' 13 - 1 ' 17 + 1 ' 



(2) 



1 11 + 1 13 — 1 17 + 1 2 



where, as in the previous product, the sign in the denominator 

 of each fraction depends on the form of the prime to which it 

 corresponds (being + for primes of the form 6w - 1, and - for 

 primes of the form 6« + 1). 



1 It follows from the first of these theorems that with the understanding 

 that no denominator is to exceed n (an indefinitely great number), 

 (1 + i)d + 4)(i + i\)(i + tS) • • • • bears a finite ratio to (1 + l)(i + -h) 

 (1 + -fa) . . . . so that as their product is known to be infinite, each of these 

 two partial products must be separately infinite ; in like manner from Euler's 

 second theorem a similar conclusion may be inferred in regard to each of the 

 two products (1 + *)(i + A) (1 + •&) (1 + »'-,) (1 + A)(i + A) • • • -and 

 (i + »(i + A)(i + A)(i + irt). • • • 



