Cyclic number

A cyclic number is an integer with a specific number of digits such that the products of this number by the integers 1 through the number of digits yield cyclic permutations of its own digits.^[1] The most prominent example is 142857, a six-digit cyclic number derived from the repeating decimal expansion of 1/7 = 0.142857142857..., where multiples from 1×142857 to 6×142857 produce rearrangements like 142857, 285714, 428571, 571428, 714285, and 857142, respectively.^[1] These numbers arise from the period of the decimal expansion of reciprocals of full reptend primes, which are primes p where the decimal period of 1/p is exactly p-1 digits long; examples include 7, 17, 19, 23, 29, 47, 59, 61, and 97.^[1] For such a prime p with period n = p-1, the cyclic number is the n-digit repeating block from the decimal expansion of 1/p, treated as a digit string.^[1] Notably, 142857 × 7 = 999999, illustrating Midy's theorem, which states that for a cyclic number generated by a full reptend prime, multiplication by p yields a string of n 9's.^[1] Cyclic numbers hold significance in number theory due to their connection to the distribution of full reptend primes among all primes, approximated by Artin's constant (approximately 0.3739558136), suggesting that about 37.4% of primes are full reptend primes, though this remains unproven for infinitude.^[1] Their study intersects with topics like primitive prime factors and cyclotomic polynomials.^[1]

Fundamentals

Definition

A cyclic number is an integer represented by a digit string of exactly d digits (possibly including leading zeros in the string representation) such that, when multiplied by any integer k where $1 \leq k \leq d, the product k \cdot c, when represented as a d-digit string (padding with leading zeros if necessary to maintain d digits and without additional digits beyond d), has digits that form a rotation of the digit string of c. This ensures no carry-over occurs during the multiplication in the cyclic sense and that the digit strings match exactly without leading zeros in the numerical value where applicable, but allowing padding for consistency.^[1]^[2] Mathematically, if c is a cyclic number with d digits, then for each k = 1, 2, \dots, d,

k \cdot c \equiv \rotate(c, s(k)) \pmod{10^d - 1},

where \rotate(c, s(k)) denotes the digit string obtained by rotating the digits of c left by s(k) positions, with s(k) depending on k, and the result is taken modulo $10^d - 1 to fit the d-digit repetend.^[1] This property holds in a given base (typically base 10), and the multiples remain permutations—specifically rotations—of the original digits in the fixed-length string.^[2] Such numbers often arise from the repeating portions of decimal expansions of reciprocals $1/n, where d divides the period length of the expansion, particularly for full reptend primes n = d + 1.^[1]

Examples

Trivial examples include the single-digit numbers 1 through 9, where the only multiple (k=1) is the number itself, forming a trivial rotation.^[1] A prominent example of a cyclic number is the six-digit integer 142857.^[1]
This sequence appears in the repeating decimal expansion of \frac{1}{7}.^[1]
When 142857 is multiplied by each integer from 1 to 6, the result is a cyclic permutation of its digits, demonstrating the defining rotational property.^[1] The multiples are as follows:

Multiplier	Product
1	142857
2	285714
3	428571
4	571428
5	714285
6	857142

Another cyclic number arises from the prime 17, forming the 16-digit sequence 0588235294117647.^[1]
Multiplying this number by integers from 1 to 16 yields rotations of its digit sequence.^[1] For the number 142857, multiplication by 7 produces 999999, a repetition of six nines that aligns with the period length but does not permute the original digits.^[3]

Relation to Repeating Decimals

Connection to fractional expansions

Cyclic numbers arise prominently in the decimal expansions of unit fractions $1/p, where p is a prime number not equal to 2 or 5, and the expansion has a maximal repeating period of exactly p-1 digits.^[1] In such cases, the repeating block of digits, known as the repetend, forms a cyclic number, meaning that the multiples of this block up to p-1 times produce cyclic permutations of its digits.^[4] This connection stems from the purely periodic nature of these expansions, where the decimal immediately enters its repeating cycle without any non-repeating digits.^[5] The mechanism behind this phenomenon is revealed through the long division process for computing $1/p. Starting with the initial remainder of 1, each subsequent step multiplies the current remainder by 10 and divides by p to obtain the next digit, with the new remainder being $10 \times (previous remainder) modulo p.^[4] When the multiplicative order of 10 modulo p—the smallest positive integer k such that $10^k \equiv 1 \pmod{p}—equals p-1, the remainders cycle through all integers from 1 to p-1 exactly once before repeating.^[5] The sequence of digits generated from these remainders, when concatenated, yields the cyclic number, and the cyclic permutation property in multiples follows because multiplying by integers from 1 to p-1 simply shifts the starting point in this full cycle of remainders.^[6] This link has been observed historically in specific examples, such as the decimal expansion of $1/7 = 0.\overline{142857}, where the 6-digit repetend 142857 is the smallest cyclic number, and its multiples (e.g., $2 \times 142857 = 285714) are rotations of the same digits.^[1] For the repetend to produce a cyclic number, the period must be the full length p-1, rather than a proper divisor of p-1, which occurs precisely when 10 is a primitive root modulo p.^[4] If the period is shorter, the expansion repeats a smaller block, and the resulting number lacks the full cyclic permutation property across all multiples up to p-1.^[5]

Full reptend primes

A full reptend prime, also known as a long prime, is a prime number p for which the decimal expansion of \frac{1}{p} has a repeating period of exactly p-1 digits, resulting in a repetend that forms a cyclic number.^[7] This maximal period length distinguishes these primes from others with shorter repeating cycles in their reciprocals.^[8] The smallest such prime is 7, where \frac{1}{7} = 0.\overline{142857} and the 6-digit repetend cycles through all nonzero residues modulo 7.^[7] The first fifteen full reptend primes are 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, and 179.^[9] This property holds if and only if 10 is a primitive root modulo p, meaning the multiplicative order of 10 modulo p is precisely p-1.^[7] In other words, the powers of 10 modulo p generate all nonzero residues in the multiplicative group modulo p.^[8] Whether there are infinitely many full reptend primes remains an open question, though heuristics indicate this is likely the case.^[10] This infinitude is a special instance of Artin's conjecture on primitive roots, which posits that for any integer a neither equal to -1 nor a perfect square, there are infinitely many primes p such that a is a primitive root modulo p; here, a = 10.^[10] The conjecture, originally proposed by Emil Artin in 1927, has been partially resolved under the generalized Riemann hypothesis but remains unproven unconditionally.

Construction and Form

Generating cyclic numbers

Cyclic numbers are constructed from full reptend primes, which are primes p for which the decimal expansion of $1/p has a maximal period length of p-1 digits.^[7] The primary method to generate the cyclic number associated with such a prime p involves performing the long division of 1 by p and extracting the first p-1 digits of the repeating decimal portion, or repetend.^[1] This process directly yields the sequence of digits that forms the cyclic number, as the full reptend ensures the period covers exactly p-1 digits before repeating.^[8] An alternative algorithmic approach simulates the long division process more explicitly: begin with an initial remainder of 1, then iteratively multiply the current remainder by 10, divide by p to obtain the next digit as the integer quotient, and take the new remainder as the result of $10 \times previous remainder modulo p; continue recording digits until the remainder returns to 1, producing the p-1 digits of the cyclic number.^[1] For example, with p = [7](/page/+7), start with remainder 1:
$10 \div [7](/page/+7) = [1](/page/1) (digit 1), remainder $10 - [7](/page/+7) = [3](/page/3);
$30 \div [7](/page/+7) = 4 (digit 4), remainder $30 - 28 = 2;
$20 \div [7](/page/+7) = 2 (digit 2), remainder $20 - 14 = 6;
$60 \div [7](/page/+7) = 8 (digit 8), remainder $60 - 56 = 4;
$40 \div [7](/page/+7) = 5 (digit 5), remainder $40 - 35 = 5;
$50 \div [7](/page/+7) = [7](/page/+7) (digit 7), remainder $50 - 49 = [1](/page/1).
The digits collected are 142857, the 6-digit cyclic number for p = [7](/page/+7).^[1] In cases producing even-length repetends, such as for p = 17, the cyclic number may begin with a leading zero to ensure it consists of exactly p-1 = 16 digits, as seen in the repetend 0588235294117647 from $1/17.^[1] This preserves the full structure required for the number's cyclic properties under multiplication.^[8]

Algebraic representation

A cyclic number c corresponding to a full reptend prime p (also known as a long prime) in base 10 is given by the closed-form expression
c = \frac{10^{p-1} - 1}{p},
where this yields an integer whose decimal representation, padded with leading zeros if necessary, consists of exactly p-1 digits that form the repeating sequence in the decimal expansion of $1/p. The formula derives from the fact that the period of the decimal expansion of $1/p is p-1, the multiplicative order of 10 modulo p, implying $10^{p-1} \equiv 1 \pmod{p} and thus p divides $10^{p-1} - 1. This ensures c is an integer, and its digits (with leading zeros if the integer has fewer than p-1 digits) represent the full repetend. For instance, with p = [7](/page/+7),
c = \frac{10^{6} - 1}{7} = \frac{999999}{7} = 142857,
which matches the repeating block in $1/7 = 0.\overline{142857}. In this representation, c captures the core algebraic structure underlying the cyclic permutations observed in multiples of the number.^[1]

Properties

Cyclic permutations in multiples

A defining property of cyclic numbers is that their multiples by integers k from 1 to d, where d is the number of digits, yield numbers that are cyclic permutations of the original digits. Specifically, for a cyclic number c associated with a full reptend prime p (where d = p-1), the multiple k \cdot c equals c rotated left by (k-1) positions in some cases, but more generally by a shift amount determined by the structure, considered modulo the digit length to preserve the permutation. This permutation arises because c = (10^d - 1)/p, so k \cdot c = k \cdot (10^d - 1)/p. Since k < p, k \cdot c < 10^d - 1, and the result is a d-digit number whose digits are a rearrangement of those in c. The mathematical basis relies on 10 being a primitive root modulo p, ensuring the multiplicative order of 10 modulo p is exactly d = p-1, which generates a full cycle of residues.^[11] The precise relation is captured by the congruence k \cdot c \equiv 10^{s(k)} \cdot c \pmod{10^d - 1}, where s(k) is the shift amount, specifically the discrete logarithm of k base 10 modulo p (i.e., the smallest nonnegative integer s such that $10^s \equiv k \pmod{p}). This equation holds because multiplying by $10^{s(k)} modulo $10^d - 1 corresponds to a left cyclic shift of the digits of c by s(k) positions, and the primitive root property guarantees that such an s(k) exists and is unique for each k = 1 to p-1. The underlying reason this works stems from the long division algorithm for decimals: the digits of $1/p are produced by successive remainders starting from 1, multiplied by 10 modulo p. For k/p, the process starts with remainder k, and since the remainders for $1/p cycle through all nonzero residues modulo p exactly once (due to the primitive root condition), the sequence for k/p is merely a cyclic shift of the original remainder sequence by the position where k first appears. Consequently, the digit sequence, determined by floor(10 \cdot remainder / p), shifts accordingly, producing the permutation in the multiples of c. For illustration, the multiples of the cyclic number 142857 (for p=7) demonstrate this shifting pattern.^[11]

Additional characteristics

Cyclic numbers exhibit notable divisibility properties tied to their generating full reptend primes. For the cyclic number 142857 associated with the prime 7, multiplication by 7 yields 999999 = $10^6 - 1, a string of six 9's, illustrating the general property.^[1] In general, a cyclic number c of period d (where d = p-1 for prime p) satisfies c \times p = 10^d - 1.^[12] A form of self-similarity arises through Midy's theorem, applicable when the period length is even. For 142857, splitting the digits into halves (142 and 857) results in their sum equaling 999, a pattern that holds for the repeating decimal expansion of $1/7. Similar truncations or substrings, such as considering 0142857 as a 7-digit sequence, do not produce cyclic behavior, as the property requires the exact period length without leading zeros or extensions.^[1] Each full reptend prime generates a unique cyclic number in base 10, with examples including 142857 from 7, 0588235294117647 from 17, and 052631578947368421 from 19. No composite numbers are known to produce cyclic numbers in base 10, as the maximal period condition typically aligns with primality.^[9] The proportion of such primes is conjectured to approach Artin's constant, approximately 0.3739558136, suggesting an infinite but sparse set of cyclic numbers.^[7] Some cyclic numbers are factors of numbers of the form $10^d - 1 in intriguing ways; for instance, $10^6 - 1 = 142857 \times 7, connecting to broader algebraic structures without direct ties to Fermat primes.^[12]

Extensions

In other number bases

The concept of cyclic numbers generalizes to arbitrary bases b > 1. In base b, a cyclic number is an integer with p-1 digits, where p is an odd prime not dividing b, such that multiplication by the integers 1 through p-1 yields cyclic permutations of its digits. This property arises when the repetend of the fraction $1/p in base b has maximal length p-1, which occurs if and only if b is a primitive root modulo p.^[13] The construction mirrors that in base 10: the cyclic number is the integer formed by the repeating block of digits in the base-b expansion of $1/p. For multiples k/p with $1 \leq k < p, the repetends are cyclic shifts of this block, ensuring the permutation property. This requires the multiplicative order of b modulo p to be exactly p-1, confirming b's status as a primitive root.^[13]^[14] Examples illustrate this in non-decimal bases. In base 3, with p=7 (where 3 is a primitive root modulo 7), the expansion of $1/7 is $0.\overline{010212}_3, and the repetend 010212 forms the cyclic number; its multiples by 1 through 6 produce rotations like 102120, 021201, and so on.^[13] Similarly, in base 13 with p=11 (13 primitive modulo 11), the repetend of $1/11 is the 10-digit sequence 12495A8973 (where A=10), and multiples yield cyclic permutations thereof, forming a 10×10 Latin square of symbols.^[15] For any fixed odd prime p, there are infinitely many bases b in which p generates a b-cyclic number, since b is a primitive root modulo p if its residue class is one of the \phi(p-1) primitive roots, and there are infinitely many such b.^[13] Trivial cases exist in small bases; for instance, in base 2 with p=3, the repetend of $1/3 = 0.\overline{01}_2 is 01, and $2/3 = 0.\overline{10}_2 is its rotation.^[16] Cyclic numbers are connected to repunits R_d = \frac{b^d - 1}{b - 1}, where d is the length of the cyclic number, through the formula for the cyclic number c = \frac{b^d - 1}{p}, linking them via the repetend period.^[1] This relation ties cyclic numbers to the theory of primitive roots modulo primes. A prime p yields a cyclic number of length p-1 in base 10 if and only if 10 is a primitive root modulo p, meaning the multiplicative order of 10 modulo p is exactly p-1, which produces a full reptend period in the decimal expansion of $1/p.^[7] Such primes are known as full reptend primes, and their existence and density are linked to Artin's conjecture on primitive roots, which posits that for any integer a not -1 or a perfect square, a is a primitive root modulo infinitely many primes, with an asymptotic density of approximately 37% for fixed a like 10.^[8] This unsolved conjecture implies infinitely many cyclic numbers in base 10, highlighting an open problem in analytic number theory regarding the distribution of such primes.^[17] Generalizations of cyclic numbers primarily occur for primes with full reptend periods; for composite n, the period is shorter than n-1, so full cyclic permutation properties do not hold, though partial cyclic behaviors may appear in certain cases.^[12] The cyclic permutation property of these numbers resonates with concepts in combinatorics, such as necklaces invariant under rotation, and in coding theory, where cyclic codes are invariant under cyclic shifts of codewords, aiding in error correction.^[18]