‘ PERFECT ’ AND £ FRIENDLY ’ NUMBERS 
75 
fifth, 2 12 (2 13 -1) = 33 550 336 
sixth, 2 1G (2 17 — 1) = 8 589 869 056 
seventh, 2 18 (2 19 —1) = 137 438 691 328 
eighth, 2 30 ( 2 31 — 1) = 2 3 0 5 8 4 3 0 0 8 1 3 9 9 5 2 1 28 
ninth, 2 G0 (2 61 — 1) = 2 658 455 991 569 831 744 654 692 
615 953 842 176 
tenth, 2 88 ( 2 89 — 1). 
With these ‘perfect’ numbers should be compared the so- 
called ‘ friendly numbers Two numbers are ‘ friendly ’ when 
each is the sum of all the aliquot parts of the other, e.g. 284 and 
220 (for 284 = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44-1-55 + 110, 
while 220 = 1 +2 + 4 + 71 + 142). lamblichus attributes the 
discovery of such numbers to Pythagoras himself, who, being 
asked ‘ what is a friend ? ’ said ‘Alter ego ’, and on this analogy 
applied the term ‘ friendly ’ to two numbers the aliquot parts 
of either of which make up the other. 1 
While for Euclid, Theon of Smyrna, and the Neo-Pytha 
goreans the ‘ perfect ’ number was the kind of number above 
described, we are told that the Pythagoreans made 10 the 
perfect number. Aristotle says that this was because they 
found within it such things as the void, proportion, oddness, 
and so on. 2 The reason is explained more in detail by Theon 
of Smyrna 3 and in the fragment of Speusippus. 10 is the 
sum of the numbers 1, 2, 3, 4 forming the rerpa/cruy ( : their 
greatest oath’, alternatively called the ‘principle of health’ 4 ). 
These numbers include the ratios corresponding to the musical 
intervals discovered by Pythagoras, namely 4 : 3 (the fourth), 
though he does not give it ; it was, however, known, with all its factors, 
in the fifteenth century, as appears from a tract written in German 
which was discovered by Curtze (Cod, lat. Monac. 14908). The first 
eight ‘perfect’ numbers were calculated by Jean Prestet (d. 1670); 
Fermat (1601-65) had stated, and Kuler proved, that 2 31 -1 is prime. 
The ninth perfect number was found by P. Seelhotf, Zeitsc.hr. f. Math. u. 
Physik, 1886, pp. 174 sq.) and verified by E. Lucas (Mathesis, vii, 1887, 
pp. 44-6). The tenth was found by R. E. Powers {Bull. Amer. Math. 
Soc., 1912, p. 162). 
1 Iambi, in Nicom., p. 35. 1-7. The subject of ‘friendly’ numbers 
was taken up by Euler, who discovered no less than sixty-one pairs of 
such numbers. Descartes and van Schooten had previously found three 
pairs but no more. 
3 Arist. Metaph. M. 8, 1084 a 32-4. 
3 Theon of Smyrna, p. 93. 17-94. 9 (Vorsokratiker, i 3 , pp. 3Q3-4). 
4 Lucian, Be lapsu in salutando, 5.