# Ormiston Tuples

Introduction
Ormiston tuples with minimal difference
Special Ormiston tuples
Two distinct digits
Large pair
Large probable pairs
Large differences
Titanic quintuple

### Introduction

An Ormiston pair is two consecutive prime numbers which are anagrams, i.e. contain the same decimal digits in different order. The first Ormiston pairs are (1913, 1931), (18379, 18397), (19013, 19031).
The term was introduced in Ormiston Pairs (pdf file), by Andy Edwards while working at Ormiston College. His students manually inspected prime lists to find small pairs.
MathWorld uses another name in a short entry: Rearrangement Prime Pair.

An Ormiston pair can be generalized to an Ormiston k-tuple: k consecutive primes which are anagrams.
The first Ormiston triples are (11117123, 11117213, 11117321) and (12980783, 12980837, 12980873). Robert G. Wilson may have been the first to find them in OEIS sequence A075093.
I computed the first Ormiston quadruples in 2006. The smallest is (6607882123, 6607882213, 6607882231, 6607882321). A search found 349 quadruples below 1012.
7 October 2012 Giovanni Resta reported in OEIS sequence A161160 that the smallest quintuple is (20847942560791, 20847942560917, 20847942560971, 20847942561079, 20847942561097).
11 October 2012 I found the 6-tuple (166389896360719, 166389896360791, 166389896360917, 166389896360971, 166389896361079, 166389896361097). It appears likely to be the smallest.

An admissable k-tuple is a tuple of k integers (n1, n2, ..., nk) such that for each prime p ≤ k, there exists at least one value which no ni is congruent to (modulo p). Equivalently (by the chinese remainder theorem) there exists integer N such that no N + ni has a prime factor ≤ k. Some versions of the prime k-tuple conjecture say: For any admissable k-tuple, there are infinitely many N for which all N + ni are primes.

In decimal, primes above 5 must end in digit 1, 3, 7 or 9 to avoid divisibility by 2 and 5.
Let us define an admissable Ormiston k-tuple as an admissable k-tuple where all k integers ni end in 1, 3, 7 or 9, and are digit anagrams (allowing leading 0's if they have different lengths). If a number N ends with at least as many 0's as there are digits in the largest ni, then all N + ni are also anagrams, and they end with the digits of ni. I conjecture: For any admissable Ormiston k-tuple, there are infinitely many such N for which the k anagrams N + ni are consecutive primes, producing an Ormiston k-tuple.

It is well-known that n and the sum of digits in n are congruent modulo 9 (meaning 9 divides their difference). If numbers are digit anagrams in decimal then they have the same sum of digits, so the numbers are congruent modulo 9. Combine this with the numbers being odd, and we get that all the numbers in an admissable Ormiston tuple are congruent modulo 18 (meaning 18 divides the difference between any two of them). The same holds for the primes in an Ormiston tuple.

Numbers ending in 1, 3, 7, or 9 always avoid the prime factors 2 and 5.
3 divides n if and only if 3 divides the sum of the digits in n. Rearranging digits does not change their sum, so 3 does not divide any anagram of a prime above 3.
In a tuple of odd anagrams not ending in 5, the smallest prime which can prevent it from being an admissable Ormiston tuple is 7.
Exhaustive search shows the minimal difference d between the smallest and largest number in an admissable Ormiston k-tuple for k = 2 to 11 is: d = 18, 90, 180, 306, 378, 900, 918, 1134, 1368, 1836.

Examples (not the only) of admissable Ormiston k-tuples with the minimal difference d:
k=2, d=18: (13, 31)
k=3, d=90: (783, 837, 873)
k=4, d=180: (0917, 0971, 1079, 1097)
k=5, d=306: (0791, 0917, 0971, 1079, 1097)
k=6, d=378: (0719, 0791, 0917, 0971, 1079, 1097)
k=7, d=900: (7813, 7831, 8137, 8173, 8317, 8371, 8713)
k=8, d=918: (7813, 7831, 8137, 8173, 8317, 8371, 8713, 8731)
k=9, d=1134: (7839, 7893, 7983, 8379, 8397, 8739, 8793, 8937, 8973)
k=10, d=1368: (1923, 2139, 2193, 2319, 2391, 2913, 2931, 3129, 3219, 3291)

The following two k-tuples are not admissable modulo 7 (meaning that 7 will divide at least one of N + ni for any N):
k=11, d=1800: (09137, 09173, 09317, 09371, 09713, 09731, 10379, 10397, 10739, 10793, 10937)
k=12, d=1836: (09137, 09173, 09317, 09371, 09713, 09731, 10379, 10397, 10739, 10793, 10937, 10973)

The latter of the two becomes admissable for k=11 if 09317 or 10793 is removed, e.g. removing 10793:
k=11, d=1836: (09137, 09173, 09317, 09371, 09713, 09731, 10379, 10397, 10739, 10937, 10973)

k=11 is the smallest k for which the closest k-tuple of anagrams ending in 1, 3, 7 or 9 is not admissable.

### Ormiston tuples with minimal difference

The smallest Ormiston pair (1913, 1931) has the minimal d = 18.
The smallest triple (11117123, 11117213, 11117321) has difference 198 which is surprisingly large; the next 571 triples have smaller differences. The second triple (12980783, 12980837, 12980873) has the minimal d = 90.
The smallest quadruple (6607882123, 6607882213, 6607882231, 6607882321) has non-minimal difference 198.
The seventh quadruple (31542480917, 31542480971, 31542481079, 31542481097) has the minimal d = 180
The smallest quintuple (20847942560791, 20847942560917, 20847942560971, 20847942561079, 20847942561097) has the minimal d=306.
The 6-tuple (166389896360719, 166389896360791, 166389896360917, 166389896360971, 166389896361079, 166389896361097) is the smallest with the minimal d = 378. It has not been exhaustively tested whether there are smaller 6-tuples with non-minimal difference.

The mentioned pair, triple, quadruple, quintuple and 6-tuple with minimal d have the ending digits of the above examples of admissable Ormiston tuples. We can express them:
k=2, d=18: 1900 + (13, 31).
k=3, d=90: 12980000 + (783, 837, 873).
k=4, d=180: 31542480000 + (0917, 0971, 1079, 1097).
k=5, d=306: 20847942560000 + (0791, 0917, 0971, 1079, 1097).
k=6, d=378: 166389896360000 + (0719, 0791, 0917, 0971, 1079, 1097).

Finding the smallest Ormiston 7-tuples and longer looks hard. It is limited how closely together a sequence of anagrams can be, so a lot of other numbers between them have to be composite. When k increases it actually becomes easier to find Ormiston k-tuples with large than with small primes, because large numbers are more likely to be composite. This is similar to k consecutive primes in arithmetic progression, where large examples but not the smallest are known for k = 7, 8, 9, 10.

The following large Ormiston 7- to 9-tuples all have the minimal difference and ending digits of the above examples.

Let N = 27853205133922751374230491248074151534244472231210475955696928877237978891600000
80-digit 7-tuple, d=900: N + (7813, 7831, 8137, 8173, 8317, 8371, 8713)
80-digit 8-tuple, d=918: N + (7813, 7831, 8137, 8173, 8317, 8371, 8713, 8731)
Note that the 8-tuple contains the 7-tuple.
However, a 9-tuple with minimal d=1134 cannot contain an 8-tuple with the minimal difference 918. It can only contain 8-tuples with a non-minimal difference ≥ 1080.

74-digit 9-tuple, d=1134: 26460346024426922096587598498580390201381951306930145595901871467050710000 + (7839, 7893, 7983, 8379, 8397, 8739, 8793, 8937, 8973)

These 7- to 9-tuples were found in 2006 with my prime constellation sieve written in C. It used selected modular equations to ensure a small prime factor in unusually many numbers among the required composites. The GMP library made prp (probable prime) tests. PARI/GP proved the primes.
29 Ormiston 8-tuples (without minimal difference) were found during the 9-tuple search.

### Special Ormiston tuples

Some computational results follow.

Two distinct digits
Edwards' original article asked for the smallest Ormiston pair using only two distinct digits. It is (1333313, 1333331).
The smallest triple is 1311111133113111000 + (133, 313, 331).

Large pair
Let p = (521410567À28851+4)À(28851−4)−1. Then (p, p+18) = (35494...83279, 35494...83297) is a 5338-digit Ormiston pair.
The primes were found and proved by PrimeForm/GW. Many prp tests were needed but the primality proofs only took seconds, because p was chosen so 28849 divides p+17, and the known prime (28849−1)/(52368383À15264764469472455023) divides p+1. If enough of the prime factorization of M−1 or M+1 is known, then PrimeForm/GW can quickly prove whether M is prime. Otherwise a more general and far slower primality proving program may be needed.

Large probable pairs
Let p = 1383603831À298305+1. p is a 29602-digit prime found by David Underbakke and Phil Carmody in 2001 during a twin prime search. I located it among thousands of large primes in The Prime Pages database, maintained by Chris Caldwell. p+1260 is the next probable prime, found with PrimeForm/GW. (p, p+1260) = (78475...78593, 78475...79853) are anagrams, so it is probably an Ormiston pair. Only little of the prime factorization of p+1259 and p+1261 is known, so there is no feasible way to prove whether p+1260 is really prime.

Let q = 1081300689À298305+1. (q−1440, q) = (61329...84609, 61329...86049) is a similar probable Ormiston pair where only q has been proven prime.

Large differences
```Let p  =  52555725910882618861120484315030042860328662480447\
631832363520558769221133607887211360255496225450939300988994\
684265609641841325052720074918193211214443383418209616598413\
275259415072329023047952904366205228850346795006128579112145\
147251732630526973817614132179879643082371067396323100788759\
647844347768582331043747814613053498332579226294079460289426\
028327922143240715371096829697967653619416529806004002671665\
898843003663403464806214749696616497726850604739822019811706\
142352554092190333831413500357190745652690836927619333533300\
560339909471401898441810950382335426199881681983164992364497\
773699023787281077376280110851583616934749688402544787995918\
513504951626060592585025644908900255369230567773920256644486\
627163823194854730741925372375036041954553090879952965379173\
129270664550880090442565894016774169916741788108243090530666\
141975694025308465795498504695113834951670729314409503261488\
552584000735324080421534802851939790012002859454316216257305\
134656063129127350675201217695544868484987186303787745300831\
510651918373099549370567725495727920574731290819747663004119  ```
p has no short expression. (p, p+37782) = (52555...04119, 52555...41901) is an Ormiston pair with a large difference. The 1070-digit primes were proved by Marcel Martin's Primo. Torbj÷rn Alm and I found the prime gap of 37782 in 2004. The gap is around 15 times larger than the average log(p) for that prime size. We were only searching large prime gaps at the time. A later test showed a few of them were Ormiston pairs. There were also 3 pairs with larger probable primes and differences 39636, 39600, 39330.

Titanic quintuple
```Let n=525757495441803442717492498764034464327799121693439999\
138711693468834684694811733417824869323091440096985981362311\
458246963658448487473078458295925908243895767856758544353346\
362923506931855823115355157377176264540764766503062427771069\
279577271660654563558693962129533233232851087980362856522539\
938656745729079339380601951512380425166013538717552146518683\
513420966988218008083603399237574640423676806512698083756364\
813030099395037557165813133280246144358196990053540646287067\
918011777810796047420110334151184452207601592281843241059001\
825391520875918137630090548062577447833570253212683867379248\
949172566196920324525508874469214290061133655530041533233660\
590559543093985894984380910965097723480094683533560369577344\
269783477009393941539692795269486356722753522850092070822595\
801801333439611475653867843619726106490963093023847315560025\
477169376164365042536688559994140155058094946223202957075832\
611993276199866424759164594817727294720307633597480289711815\
496991872281261569527767888721253039470089461149792442700000 ```
n + (19723, 21973, 27139, 27319, 29713) is a 1014-digit Ormiston quintuple. n has no short expression. Its modular properties were chosen to efficiently search a titanic (meaning at least 1000 digits) quintuple. This required 5 out of a longer sequence of anagrams to be primes, while most of the numbers between them were guaranteed to have a small prime factor. PrimeForm/GW and the GMP library made prp tests. Primo proved the primes.

Made by Jens Kruse Andersen, jens.k.a@get2net.dk   home
I would like to hear of interesting results (not in other bases than decimal).
Page created 28 April 2007. Last updated 11 October 2012.