完全數
完全數者，正整數也。其正因子之和適等於本數。如六，一、二、三之和也；二十八，一、二、四、七、十四之和也。
完全數者，正整數也。其正因子之和適等於本數。如六，一、二、三之和也；二十八，一、二、四、七、十四之和也。[一]
昔歐幾里得證之：二冪減一為素數，則此素數與前冪之積為偶完全數。如六、二十八、四百九十六、八千一百二十八是也。[二]後歐拉復證：偶完全數必如是形。[三]
迄今未知奇完全數之存否，亦未明完全數之多寡。[四]然學者考之，得若干性質：
其二，偶完全數之二進式表甚特，皆若干一後接等數之零。[六]
其三，完全數之因子之倒數和必為二。[七]
其四，完全數之因子數必為偶。[八]
奇完全數若存，必大於十之千五百冪。其因子至少百有一。[九]
數學家窮索數千載，然此數之奧秘尚未盡明。非淺學所能窺其堂奧，唯精於此道者，方能探其精微。[一〇]
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纂- ↑ Dickson, L. E. (1919). History of the Theory of Numbers, Vol. I. Washington: Carnegie Institution of Washington. p. 4.
- ↑ Caldwell, Chris, "A proof that all even perfect numbers are a power of two times a Mersenne prime".
- ↑ Euler, Leonhard. "De numeris amicabilibus". Archived from the original on 2019-04-14.
- ↑ Pomerance, Carl (1974). "Odd perfect numbers are divisible by at least seven distinct primes". Acta Arithmetica. 25 (3): 265–300.
- ↑ Luo, Ming (2014). "Triangular numbers and their sums". arXiv:1408.5755 [math.NT].
- ↑ Holdener, Judy A. (2002). "A Theorem of Touchard on the Form of Odd Perfect Numbers". The American Mathematical Monthly. 109 (7): 661–663.
- ↑ Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. ISBN 0-387-20860-7.
- ↑ Cohen, G. L.; Hagis, P. Jr. (1998). "Every positive integer ≥ 2 is the sum of three squares of deficient numbers". Mathematics of Computation. 67 (224): 1745–1749.
- ↑ Ochem, Pascal; Rao, Michaël (2012). "Odd perfect numbers are greater than 10^1500". Mathematics of Computation. 81 (279): 1869–1877.
- ↑ Rosen, Kenneth H. (2011). Elementary Number Theory and Its Applications (6th ed.). Addison-Wesley. ISBN 978-0-321-50031-2.