74 
PYTHAGOREAN ARITHMETIC 
further objection that (h) and (c) overlap, in fact (h) includes 
the whole of (c). 
‘ Perfect * and ‘ Friendly ’ numbers. 
There is no trace in the fragments of Philolaus, in Plato or 
Aristotle, or anywhere before Euclid, of the perfect number 
(reXeios) iu the well-known sense of Euclid’s definition 
(YII. Def. 22), a number, namely, which is ‘ equal to (the 
sum of) its own parts’ (i.e. all its factors including 1), 
e.g. 6 = 1 + 2 + 3 ; 28 = 1+2 + 4 + 7 + 14; 
496 = 1 +2 + 4 + 8 + 16 + 31 + 62 + 124 + 248. 
The law of the formation of these numbers is proved in 
Eucl. IX. 36, which is to the effect that, if the sum of any 
number of terms of the series 1, 2, 2 2 , 2 3 .... 2 n ~ l (= S n ) is prime, 
then S n . 2 n ~ 1 is a £ perfect ’ number. Theon of Smyrna 1 and 
Nicomachus 2 both define a ‘perfect’ number and explain the 
law of its formation ; they further distinguish from it two 
other kinds of numbers, (1) over-perfect {imepreXiis or vTrepri- 
Aeioy), so called because the sum of all its aliquot parts is 
greater than the number itself, e.g. 12, which is less than 
1 + 2 +3 + 4+ 6, (2) defective {¡XXnnfc), so called because the 
sum of all its aliquot parts is less than the number itself, 
e. g. 8, which is greater than 1+2 + 4. Of perfect numbers 
Nicomachus knew four (namely 6, 28, 496, 8128) but no more. 
He says they are formed in ‘ ordered ’ fashion, there being one 
among the units (i.e. less than 10), one among the tens (less 
than 100), one among the hundreds (less than 1000), and one 
among the thousands (less than a myriad); he adds that they 
terminate alternately in 6 or 8. They do all terminate in 6 or 
8 (as we can easily prove by means of the formula (2 n — 1) 2 n ~ l ), 
but not alternately, for the fifth and sixth perfect numbers 
both end in 6, and the seventh and eighth both end in 8. 
lamblichus adds a tentative suggestion that there may (e/ 
rayoi) in like manner be one perfect number among the first 
myriads (less than 10000 2 ), one among the second myriads 
(less than 10000 3 ), and so on ad infinitum. 3 This is incorrect, 
for the next perfect numbers are as follows : 4 
1 Theon of Smyrna, p. 45. 2 Nicom. i. 16, 1-4. 
3 Iambi, in Nicom., p. 33. 20-28. 
4 The fifth perfect number may have been known to lamblichus,