# What Are The Perfect Numbers From 1 To 100?

6, 28, 496, 8128.

A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself).

The first four perfect numbers are 6, 28, 496 and 8128. Much later five more were found: 33 550 336 552 784 969 152 930 438 048 176 640 000 (sequence A000396 in OEIS). As of January 2016 there are 49 known odd perfect numbers (sequence A001158 in OEIS) and four known even ones (sequence A001220), with the largest recent discovery being L(377860871787183407)=2^155Â·3^255Â·5^357Â·7^167Â·11^271Â·13^172Â·17nbsp;- 1 with n = 2956156169297611206464016654240128512507587586274073076598574692736860803840000000000000000000000000000 (~ 2 Ã— 1046) discovered by James Gimbel et al.. No one knows if there are infinitely many perfect numbers or not. It has been shown however that if someone could prove that there are infinitely many prime numbers then it would follow from Euclid's proof for infinitude of primes that there also exist infinitely manyperfect numbers. Euclid's argument goes as follows: suppose towards contradictionthat p1,...pn be all prime integers greater than some given integer N0 . Then considerthe productN0(p1p2...pn)+1It must have at least one prime factor q different from any pi , since otherwise wewould haveq|N0orq|(p1p2...pn),which contradicts our assumption about q being different from each pi . So thisproduct must be composite and hence has a proper divisor d < N0(p1 p2 ... pn)+ 1<=(N02+ 2N02+ 3N02++nn)(pp...pp)=NN++=NN+(nâˆ’12)(pp)...+(nâˆ’12)(pp)>NNsince pp > pp for n â‰¥ 3 . This contradicts our initial assumption aboutthere only existing finitely many primes larger than N0 , thus proving theirinfinite existence by reductio ad absurdum."

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