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In mathematics, a Lucas–Carmichael number is a positive composite integer ''n'' such that if ''p'' is a prime factor of ''n'', then ''p'' + 1 is a factor of ''n'' + 1. They are named after Édouard Lucas and Robert Carmichael. By convention, a number is only regarded as a Lucas–Carmichael number if it is odd and square-free (not divisible by the square of a prime number), otherwise any cube of a prime number, such as 8 or 27, would be a Lucas–Carmichael number (since ''n''3 + 1 = (''n'' + 1)(''n''2 − ''n'' + 1) is always divisible by ''n'' + 1). Thus the smallest such number is 399 = 3 × 7 × 19; 399+1 = 400; 3+1, 7+1 and 19+1 are all factors of 400. The first few numbers, and their factors, are : The smallest Lucas-Carmichael number with 5 factors is 588455 = 5 × 7 × 17 × 23 × 43. It is not known whether any Lucas–Carmichael number is also a Carmichael number. ==References== * ''Unsolved Problems in Number Theory'' (3rd edition) by Richard Guy (Springer Verlag, 2004), section A13. * (PlanetMath ) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lucas–Carmichael number」の詳細全文を読む スポンサード リンク
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