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In number theory, a full reptend prime, full repetend prime, proper prime〔Dickson, Leonard E., 1952, ''History of the Theory of Numbers, Volume 1'', Chelsea Public. Co.〕 or long prime in base ''b'' is a prime number ''p'' such that the formula : (where ''p'' does not divide ''b'') gives a cyclic number. Therefore the digital expansion of in base ''b'' repeats the digits of the corresponding cyclic number infinitely, as does that of with rotation of the digits for any ''a'' between 1 and ''p'' - 1. The cyclic number corresponding to prime ''p'' will possess ''p'' - 1 digits if and only if ''p'' is a full reptend prime. That is, ord''b''''p'' = ''p'' - 1. Base 10 may be assumed if no base is specified, in which case the expansion of the number is called a repeating decimal. In base 10, if a full reptend prime ends in the digit 1, then each digit 0, 1, ..., 9 appears in the repetend the same number of times as each other digit.〔 The values of ''p'' less than 1000 for which this formula produces cyclic numbers in decimal are: :7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593, 619, 647, 659, 701, 709, 727, 743, 811, 821, 823, 857, 863, 887, 937, 941, 953, 971, 977, 983, ... For example, the case ''b'' = 10, ''p'' = 7 gives the cyclic number 142857; thus 7 is a full reptend prime. Furthermore, 1 divided by 7 written out in base 10 is 0.142857 142857 142857 142857... Not all values of ''p'' will yield a cyclic number using this formula; for example ''p'' = 13 gives 076923 076923. These failed cases will always contain a repetition of digits (possibly several) over the course of ''p'' - 1 digits. The known pattern to this sequence comes from algebraic number theory, specifically, this sequence is the set of primes p such that 10 is a primitive root modulo p. Artin's conjecture on primitive roots is that this sequence contains 37.395..% of the primes. The term "long prime" was used by John Conway and Richard Guy in their ''Book of Numbers''. Confusingly, Sloane's OEIS refers to these primes as "cyclic numbers." ==Patterns of occurrence of full reptend primes== Advanced modular arithmetic can show that any prime of the following forms: #40''k''+1 #40''k''+3 #40''k''+9 #40''k''+13 #40''k''+27 #40''k''+31 #40''k''+37 #40''k''+39 can ''never'' be a full reptend prime in base-10. The first primes of these forms, with their periods, are: However, studies show that ''two-thirds'' of primes of the form 40''k''+''n'', where ''n'' ≠ are full reptend primes. For some sequences, the preponderance of full reptend primes is much greater. For instance, 285 of the 295 primes of form 120''k''+23 below 100000 are full reptend primes, with 20903 being the first that is not full reptend. 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「Full reptend prime」の詳細全文を読む スポンサード リンク
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