This article is about numbers where permutations of their digits (in some base) yield related numbers. For the number theoretic concept, see cyclic number (group theory).
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are
To qualify as a cyclic number, it is required that consecutive multiples be cyclic permutations. Thus, the number 076923 would not be considered a cyclic number, because even though all cyclic permutations are multiples, they are not consecutive integer multiples:
076923 × 1 = 076923
076923 × 3 = 230769
076923 × 4 = 307692
076923 × 9 = 692307
076923 × 10 = 769230
076923 × 12 = 923076
The following trivial cases are typically excluded:
single digits, e.g.: 5
repeated digits, e.g.: 555
repeated cyclic numbers, e.g.: 142857142857
If leading zeros are not permitted on numerals, then 142857 is the only cyclic number in decimal, due to the necessary structure given in the next section. Allowing leading zeros, the sequence of cyclic numbers begins:
For example, the case b = 10, p = 7 gives the cyclic number 142857, and the case b = 12, p = 5 gives the cyclic number 2497.
Not all values of p will yield a cyclic number using this formula; for example, the case b = 10, p = 13 gives 076923076923, and the case b = 12, p = 19 gives 076B45076B45076B45. These failed cases will always contain a repetition of digits (possibly several).
The first values of p for which this formula produces cyclic numbers in decimal (b = 10) are (sequence A001913 in the OEIS)
This procedure works by computing the digits of 1/p in base b, by long division. r is the remainder at each step, and d is the digit produced.
The step
n = n⋅b + d
serves simply to collect the digits. For computers not capable of expressing very large integers, the digits may be output or collected in another way.
If t ever exceeds p/2, then the number must be cyclic, without the need to compute the remaining digits.
Properties of cyclic numbers
When multiplied by their generating prime, the result is a sequence of b − 1 digits, where b is the base (e.g. 10 in decimal). For example, in decimal, 142857 × 7 = 999999.
When split into groups of equal length (of two, three, four, etc... digits), and the groups are added, the result is a sequence of b - 1 digits. For example, 14 + 28 + 57 = 99, 142 + 857 = 999, 1428 + 5714+ 2857 = 9999, etc. ... This is a special case of Midy's Theorem.
All cyclic numbers are divisible by b − 1 where b is the base (e.g. 9 in decimal) and the sum of the remainder is a multiple of the divisor. (This follows from the previous point.)
Other numeric bases
Using the above technique, cyclic numbers can be found in other numeric bases. (Not all of these follow the second rule (all successive multiples being cyclic permutations) listed in the Special Cases section above) In each of these cases, the digits across half the period add up to the base minus one. Thus for binary, the sum of the bits across half the period is 1; for ternary, it is 2, and so on.
In binary, the sequence of cyclic numbers begins: (sequence A001122 in the OEIS)
In ternary (b = 3), the case p = 2 yields 1 as a cyclic number. While single digits may be considered trivial cases, it may be useful for completeness of the theory to consider them only when they are generated in this way.
It can be shown that no cyclic numbers (other than trivial single digits, i.e. p = 2) exist in any numeric base which is a perfect square, that is, base 4, 9, 16, 25, etc.
A palindromic number is a number that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term palindromic is derived from palindrome, which refers to a word whose spelling is unchanged when its letters are reversed. The first 30 palindromic numbers are:
10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language.
11 (eleven) is the natural number following 10 and preceding 12. It is the first repdigit. In English, it is the smallest positive integer whose name has three syllables.
In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system. The word is a portmanteau of "repeated" and "digit". Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes.
22 (twenty-two) is the natural number following 21 and preceding 23.
27 is the natural number following 26 and preceding 28.
79 (seventy-nine) is the natural number following 78 and preceding 80.
73 (seventy-three) is the natural number following 72 and preceding 74. In English, it is the smallest natural number with twelve letters in its spelled out name.
31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number.
A permutable prime, also known as anagrammatic prime, is a prime number which, in a given base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to study these primes, called them permutable primes, but later they were also called absolute primes.
1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.
700 is the natural number following 699 and preceding 701.
The number 142,857 is a Kaprekar number.
229 is the natural number following 228 and preceding 230.
A Friedman number is an integer, which represented in a given numeral system, is the result of a non-trivial expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×, ÷), additive inverses, parentheses, exponentiation, and concatenation. Here, non-trivial means that at least one operation besides concatenation is used. Leading zeros cannot be used, since that would also result in trivial Friedman numbers, such as 024 = 20 + 4. For example, 347 is a Friedman number in the decimal numeral system, since 347 = 73 + 4. The decimal Friedman numbers are:
181 is the natural number following 180 and preceding 182.
In number theory, a full reptend prime, full repetend prime, proper prime or long prime in base b is an odd prime number p such that the Fermat quotient
A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic ; if this sequence consists only of zeros, the decimal is said to be terminating, and is not considered as repeating. It can be shown that a number is rational if and only if its decimal representation is repeating or terminating. For example, the decimal representation of 1/3 becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333.... A more complicated example is 3227/555, whose decimal becomes periodic at the second digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144.... Another example of this is 593/53, which becomes periodic after the decimal point, repeating the 13-digit pattern "1886792452830" forever, i.e. 11.18867924528301886792452830....
The digits of some specific integers permute or shift cyclically when they are multiplied by a number n. Examples are:
Gardner, Martin. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments From Scientific American. New York: The Mathematical Association of America, 1979. pp.111–122.
Kalman, Dan; 'Fractions with Cycling Digit Patterns' The College Mathematics Journal, Vol. 27, No. 2. (Mar., 1996), pp.109–115.
Leslie, John. "The Philosophy of Arithmetic: Exhibiting a Progressive View of the Theory and Practice of ....", Longman, Hurst, Rees, Orme, and Brown, 1820, ISBN1-4020-1546-1
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