Cyclic Numbers

To quote from the source of all wisdom, Wikipedia:

A cyclic number is an integer in 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 

142857 × 1 = 142857

142857 × 2 = 285714

142857 × 3 = 428571

142857 × 4 = 571428

142857 × 5 = 714285

142857 × 6 = 857142

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 

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: 

(\(10^6-1\)) ÷ 7 = 142857 (6 digits)

(\(10^{16}-1\)) ÷ 17 = 0588235294117647 (16 digits)

(\(10^{18}-1\)) ÷ 19 = 052631578947368421 (18 digits)

(\(10^{22}-1\)) ÷ 23 = 0434782608695652173913 (22 digits)

(\(10^{28}-1\)) ÷ 29 = 0344827586206896551724137931 (28 digits)

However, it was in the realm of repeating decimals that I first encountered cyclic numbers. Specifically, I was looking at OEIS entries for 25801, a number representing my diurnal age at the time of composing this blog post, and I came across this entry:

A056215: primes \(p\) for which the period of reciprocal = \( \displaystyle \frac{p-1}{10} \) 

281, 521, 1031, 1951, 2281, 2311, 2591, 3671, 5471, 5711, 6791, 7481, 8111, 8681, 8761, 9281, 9551, 10601, 11321, 12401, 13151, 13591, 14831, 14951, 15671, 16111, 16361, 18671, 21191, 21521, 21881, 24281, 24551, 25391, 25801, ...

Sure enough, when I checked on SageMathCell, the period of this reciprocal is indeed 2580 (see Figure 1):

Figure 1: permalink

After 2580 decimal places, the digits do repeat (shown in red):

0.0000387581876671446843145614511065462578969807371807294290918956629588000465098252005736211774737413278555094763768846168753149102747955505600558117902406883454129684895934266113716522615402503778923297546606720669741482888260144955621875121119336459827138483004534707957055928064803689779465912173946746250145343203751792566179605441649548467113677764427735359094608736095500174411844502151079415526529979458160536413317313282430913530483314600209294213402581295298631835975349792643695980775938917096236579977520251153056083097554358358203170419751172435176931126700515483895973024301383667299717065230029843804503701406922212317352040618580675167629161660400759660478276035812565404441688306654780822448742296810201154993992480911592573931242975078485330025967985736986938490756172241385992790977093911088717491570094182396031161582884384326188907406689663191349172512693306460989884113018875237393899461261191426688888027595829619007015231967753187860935622650284872679353513429712026665633114995542808418278361303825433122747180341847215224216115654431998759737994651370101934033564590519747296616410216658269059338785318398511685593581644122320840277508623696755939692259989922871206542382078214022712297972946785008333010348436107127630711987907445447850858493856827254757567536142009999612418123328553156854385488934537421030192628192705709081043370411999534901747994263788225262586721444905236231153831246850897252044494399441882097593116545870315104065733886283477384597496221076702453393279330258517111739855044378124878880663540172861516995465292042944071935196310220534087826053253749854656796248207433820394558350451532886322235572264640905391263904499825588155497848920584473470020541839463586682686717569086469516685399790705786597418704701368164024650207356304019224061082903763420022479748846943916902445641641796829580248827564823068873299484516104026975698616332700282934769970156195496298593077787682647959381419324832370838339599240339521723964187434595558311693345219177551257703189798845006007519088407426068757024921514669974032014263013061509243827758614007209022906088911282508429905817603968838417115615673811092593310336808650827487306693539010115886981124762606100538738808573311111972404170380992984768032246812139064377349715127320646486570287973334366885004457191581721638696174566877252819658152784775783884345568001240262005348629898065966435409480252703383589783341730940661214681601488314406418355877679159722491376303244060307740010077128793457617921785977287702027053214991666989651563892872369288012092554552149141506143172745242432463857990000387581876 ...

Now the Wikipedia article states that:

If the digital period of \( \displaystyle \frac{1}{p}\) where \(p\) is prime is \(p − 1\), then the digits represent a cyclic number.

That clearly isn't the case with 25801. However, in the comments to the OEIS entry, it states:

Cyclic numbers of the tenth degree (or tenth order): the reciprocals of these numbers belong to one of ten different cycles. Each cycle has the following number of digits: 

\( \displaystyle \frac{number- 1}{10} \)

So 25801 is not a prime that produces a cyclic number. The nearest smaller prime that does this is 25793 and the nearest larger prime that does this is 25847. The primes that produce cyclic numbers are known as reptend primes. In this regard, Wikipedia states that:

Cyclic numbers are related to the recurring digital representations of unit fractions. A cyclic number of length \(L\) is the digital representation of: 

\(\displaystyle \frac{1}{L + 1} \)

Conversely, if the digital period of \( \displaystyle \frac{1} {p} \), where \(p \) is prime, is \(p-1\), then the digits represent a cyclic number. 

For example: 

\( \displaystyle \frac{1}{7}\) = 0.142857 142857…. 

Multiples of these fractions exhibit cyclic permutation: 

\( \displaystyle \frac{1}{7}\) = 0.142857 142857…

\( \displaystyle \frac{2}{7}\) = 0.285714 285714…

\( \displaystyle \frac{3}{7}\) = 0.428571 428571…

\( \displaystyle \frac{4}{7}\) = 0.571428 571428…

\( \displaystyle \frac{5}{7}\) = 0.714285 714285…

\( \displaystyle \frac{6}{7}\) = 0.857142 857142….

Checking reveals the nearby primes to 25801, namely 25793 and 25847, do have periods of 25792 and 25846 respectively. These periods are decimal maximal periods, the maximum length that a repeating decimal of this form can have. See Figure 2.

Figure 2: permalink

I accidentally typed in 258487 when checking and it turns out that the period of its reciprocal is 258486 and so it is a reptend prime as well. However, as stated earlier, 25801 is not a reptend but is regarded as a cyclic number of the tenth degree or tenth order. The terminology can get confusing. Let's start again with the reptend primes. These are also referred to as long period primes and form part of a categorisation involving different values of \(k\):$$\text{Primes }p \text{ such that the period of }\displaystyle \frac{1}{p} \text{ is } \displaystyle \frac{p-1}{k} \text{ where }k=1,2,3,4, ...$$The sequences generated by applying these formulae are listed in the cross references for the original OEIS A056215 entry:

A006883, A097443, A055628, A056157, A056210, A056211, A056212, A056213, A056214, A056215, A056216, A056217, A098680, which are sequences of primes \(p \) where the period of the reciprocal is \( \displaystyle \frac{p-1}{k}\) for \(k=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13\) respectively.

So I think my confusion has been clarified. It is only when \(k=1\) that the reciprocal produces a cyclic number. The Wikipedia goes on to state that:

From the relation to unit fractions, it can be shown that cyclic numbers are of the form of the Fermat quotient:$$ \displaystyle \frac {b^ {\,p-1}-1}{p}$$ where \(b\) is the number base (10 for decimal), and \(p\) is a prime that does not divide \(b\). Primes \(p\) that give cyclic numbers in base \(b \) are called full reptend primes or long primes in base \(b\). 

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).

I won't go into the other number bases here and I haven't followed up on the link to Fermat primes. Here is a Numberphile YouTube video about cyclic numbers:

The video highlights how some of the formulae arise. For example:$$ \begin{align}\frac{1}{7} &=0.142857142857142857142857 ... \\
10^6 \times \frac{1}{7} &=142857.142857142857142857142857 ... \\
\frac{10^6}{7}-142857 &=0.142857142857142857142857 ...\\
\frac{10^6}{7}-142857 &=\frac{1}{7} \\
142857&=\frac{10^6-1}{7}\\
\end{align}$$Interestingly, there is even a reference to Gurdjieff in this video because the number 142857 features in his enneagram as shown in Figure 3.

Figure 3: 

As explained in Wikipedia:

The Fourth Way enneagram is a figure published in 1949 in In Search of the Miraculous by P.D. Ouspensky, and an integral part of the Fourth Way esoteric system associated with George Gurdjieff. The term "enneagram" derives from two Greek words, ennea (nine) and gramma (something written or drawn).

I won't further into that here as this blog focuses purely on Mathematics but I may follow up on it in my other blogs.

on February 27th 2021