212 



- KNO^A/'LEIDGE ♦ 



[August 1, 1889. 



that this alternate resemblance to male and female parent 

 (the eUlest always resembling the mother) might possibly 

 have a wiilei' application than the limits of this particular 

 family, and ever since that time I have taken every oppor- 

 tunity of putting the matter to the test. Though every 

 case did not exhibit the phenomenon so strikingly as the 

 first one observed (in which the great difference in the 

 mental and physical development of the male and female 

 jiarent was very marked), yet, on the whole, the results of 

 all subsequent observations have tended to confirm my 

 original impression. Perhaps you, or some of your readers 

 who take an interest in such subjects, might think the 

 matter worthy of notice.— Your obedient servant, 

 FiiANfis Heron 

 (Demonstrator R. Coll. Surgeons). 

 Eoyal College of Surgeons, Dublin : 

 July 13, 1889. 



SOME PROPERTIES OF NUMBERS. 



By Eubert W. D. Chuistie, M.A. 



(1) 



V Pollock's Theorems every odd num- 

 ber may be divided into four squares 

 in four dilferent forms, the algebraic 

 sums of whose roots may equal 1, 3, 5 

 . . . 2;; —1 up to the maximum, ej/., 

 73 = 42 _,. 42 ^ 42 + 52 ^ 22 + 22-1- 42 + 

 7- = 02-|-P-|-6»-|-62=P + 22 4- 2- 

 -i-8-. Here the sums of the roots 

 are respectively 17 (the maximum) 15, 13 and 13. By 

 changing the signs of one or more of the roots we may make 

 the sums of the roots equal to any odd number from 1 to 1 7 

 inclusive. Thus r.g. 73=( - 1)--|- 2--t-2--)-8^ and the sums 

 of the roots= 1 1 = ( - 1 ) -f 2 -f 2 -(- 8, &c. 



Lagi-ange and others have demonstrated that ])rimes of 

 the Form 4N + 1 may be di\'ided into two different integral 

 square numbers, e.g., 5 = l--(-2^; 13=:2'--|-32, itc. I have 

 extended this well known theorem as follows : — 



((•■■'-I- 1)"'(-1:N-I- 1 )=two integral squares where m and r 

 may be any integers whatever. Thus, e.g., let ??i=2, r:=2. 

 Then (2^-1- 1)2 X 29=25 X 29=7- + 2(J2. 

 And since every even square greater than 5 may be 

 divided into four 4N-I-1 primes, we can thus resolve any 

 even square into si.cteeit squares by Pollock's, or into eight 

 ditl'erent squares by Lagrange's Theorem, e.g., 18-^37-1-4:1 

 -f97 + 149 = -ll-|-Gl+73 + l-19 = ll+53-|-73-H57=41 + 

 73-|-97-M13 = 41-l-73-l-101_-fl09, &c., ic, and each of 

 these primes may be split up into four or into two different 

 squares (id libitum. Thus, e.g., 10-=:13-|- 17-1-29-1-4:1 

 = (2--i-32) + (124-42)-i-(22-|-52)-|-(4'-'-|-52) by Lagrange=16 

 .squares by Pollock, since 13, 17, 29, & 41 are odd primes. 



(2) By means of the Diojjhantine Analysis it is easy to 

 show that if N=:fi2 -|- b'\ it also equals [ ['Imnb + (n^ — m-)a] / 

 (//i^-t-?i-)]'-f-[ {2»ma-|-(»/i- — 7i^)b] j{in- + n-)Y, where m and 

 u may be assumed at pleasure. Thus, e.g., if to=2, »i=l, we 

 may say 13=2^-1- 32 = (?)^-|-(y)'-, and so on. 



By combining (1) with (2) we can thus split up any 

 prime of Form 4N-fl into as many pairs of squares as we 

 think proper. Thus, e.g'., 6-=l + 5-|-13-|-17 = 12-l-l-'-t- 2'- + 

 22 + 32 + 12 + 4-', &c., &c. 



If we wish to divide any square number into two other 

 square integers, the following is a convenient equation for 

 our purpose :— {2r''^ -I- (2« -|- l)2r + in' + 1} 2= {2r- + ■2r(2n + 

 l)-|-4/ti^4- |(2« — l)(2j--|-2n + 1)! -, where n and ;• maybe 

 any integers. Thus, e.g., if r=3, m = 2, we have 65^^ 

 56^-1-33^. It may be incidenUvlly mentioned that one of 

 the three squares is always divisible by 5. 



(3) Mainly from the simple fact that 2-|-2=2 x 2 I have 

 constructed an interesting theorem, which will enable us to 

 turn n integral squares into n other integral squares in w-|- 1 

 different ways. (nai")--|-(?iaf )--|-(jjaJ')- . . . (»iaP')-= 

 {2(a"' + a^ . . . af.j) — {n—2)a'^'}^ + n — 1 corresponding 

 symmetrical expressions, e.g. ; let « = 3 m=:l, then wc have 

 (:ia,Y + {3a.;)-' + {Sa,)^={2a,+2a.,-a,Y+{2a.^ + 2a,-a,y 

 + (2«3 -1- 2«i - «.,]^z=( - 2«i -f 2a2 -aj)- + (2a2 + 2^3 + a, )- -i- 

 (2a3-2a,-a.,)2==(2ai-2a.,-a3)2 + (-2a, + 2^3 - a,)= + 

 (2o3 + 2a, -I- a2)2— (2(Ji + 2a^ + a^f + (2a,-2a3-a,)2 + 

 (-2n5-f-2a,-«2)2. 



If integers be desired, leta|^2a2:=5 n3:=ll, say, and 

 we have at once 6- + 152 + 33'^=3--l-30- + 21-=52-|-34- + 

 13-=17--f 102 + 31-=25'-^-f-14'H23-. And since the sum 

 of the roots of the first three (or n) squares always equals 

 the sum of the second three (or n) squares, we can raise or 

 depress the equations ad libitum. Thus, e.g., instead of 

 G-'-f-15--f 33-=3--|-30- + 21-, we may add or subtract any 

 number from each member of the equation without vitiating 

 it. Thus, e.g., we may write down at once 7- -I- 16'--|-34'-= 

 4-'-f-31--H22- or 5--M4--|-32-=2--|-29--|-20-, and so on. 



(4) In order to obtain a square number equal to n other 

 si|uare numbers, we have L'^-=A;-|-Ao . . . Af„ if Ai = 



a\ + al (r„, and A;, ^ 2a,«,„ A3 = 2a2«„ ... A„ = 



2<Jn-iOn- £-'J-, suppose WC Want a square number equal to 

 seven, other square numbers : — 



Let f(|^l ((,,=3 rt3^5 «4=7 «.5 = 2 «(,=4 a„=8, say. 



Then A,=40 A„=16 A3=48 Aj=80 A5=112 Ab=32 

 A7=64. Thus, after cancellation, 21-'=5- + 2- -i- G- -|- 1 0- -|- 

 142 + 42 + 82. 



II. A perfect number is one which is equal to the sum of 

 all the lesser numbers which divide it without remainder. 

 These numbers have engaged the attention of mathe- 

 maticians from very early times. Euclid treats of them in 

 the last proposition of his Ninth Book. No odd perfect 

 number has yet been discovered, and the most eminent 

 mathematicians are at variance on the point. Descartes in 

 1G38 wrote: — "Je juge qu'on pent trouver des nombres 

 impairs veritablement parfaits." On the other hand. Pro- 

 fessor Sylvester says, December 15, 1887: — "There seems 

 every I'eason to believe that Euclid's Perfect Numbers are 

 the only perfect numters which exist." The formula for a 

 perfect number is 7'=:2"(2"+' — 1) provided 2"*' — 1 be a 

 prime, i.e., provided u + l be a (not tnij/) prime. If we 

 could solve the problem " What prime p will make 2''— 1 a 

 prime," there would be little difficulty in obtaining any 

 number of perfect numbers, but as it is, only about a dozen 

 perfect numbers are known. 



The first six perfects are 6, 28, 496, 8128, 33550336, 

 8589869056. Thus, e.g., the factors of G are 1, 2, 3, and 

 1 +2 + 3 = 6. Therefore G is a perfect number. Again, the 

 factors of 28 are 1,2,4,7, 1 4, and 1 + 2 + 4 + 7 + + 1 4 = 28. 

 Thus 28 is also a perfect number, and so on with the 

 others. From the above formula for perfects we make 

 the following deductions : — 



(1) Every perfect number equals a sequence from unity 

 but from no other number. 



(2) Perfects are all of Form 9 N -1- 1 (except the first). 



(3) They must end in the digits 6 or 8. 



(4) The product of the factors of a Perfect number is 

 an exact power of t'ne Perfect itself, i.e. Product of Factors 

 =P"={2"(2''+'-l)}". 



(5) The number of po.sitive integers less than the Perfect 

 and prime to it divided by the Perfect is less than one-half 

 or</.(P)/P<i. 



(G) Every perfect number except the fii-st equals an e.ien 

 number of odd cubes, e.g. 28^r' + 3'; also 496=l*' + 3-' 

 + 5-' + 73 ; also Product of Factors of 28= 1x2x4x7x14 

 = 784=28"- {v. No. (4) above). 



