Admissable tuples

Minimal admissable difference

Ormiston tuples with minimal difference

Special Ormiston tuples

Two distinct digits

Large pair

Large probable pairs

Large differences

Titanic quintuple

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

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 10

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.

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

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.

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

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.

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.

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).

Let p = (521410567·2

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 2

Let p = 1383603831·2

Let q = 1081300689·2

Let p = 52555725910882618861120484315030042860328662480447\ 631832363520558769221133607887211360255496225450939300988994\ 684265609641841325052720074918193211214443383418209616598413\ 275259415072329023047952904366205228850346795006128579112145\ 147251732630526973817614132179879643082371067396323100788759\ 647844347768582331043747814613053498332579226294079460289426\ 028327922143240715371096829697967653619416529806004002671665\ 898843003663403464806214749696616497726850604739822019811706\ 142352554092190333831413500357190745652690836927619333533300\ 560339909471401898441810950382335426199881681983164992364497\ 773699023787281077376280110851583616934749688402544787995918\ 513504951626060592585025644908900255369230567773920256644486\ 627163823194854730741925372375036041954553090879952965379173\ 129270664550880090442565894016774169916741788108243090530666\ 141975694025308465795498504695113834951670729314409503261488\ 552584000735324080421534802851939790012002859454316216257305\ 134656063129127350675201217695544868484987186303787745300831\ 510651918373099549370567725495727920574731290819747663004119p 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. Torbjörn 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.

Let n=525757495441803442717492498764034464327799121693439999\ 138711693468834684694811733417824869323091440096985981362311\ 458246963658448487473078458295925908243895767856758544353346\ 362923506931855823115355157377176264540764766503062427771069\ 279577271660654563558693962129533233232851087980362856522539\ 938656745729079339380601951512380425166013538717552146518683\ 513420966988218008083603399237574640423676806512698083756364\ 813030099395037557165813133280246144358196990053540646287067\ 918011777810796047420110334151184452207601592281843241059001\ 825391520875918137630090548062577447833570253212683867379248\ 949172566196920324525508874469214290061133655530041533233660\ 590559543093985894984380910965097723480094683533560369577344\ 269783477009393941539692795269486356722753522850092070822595\ 801801333439611475653867843619726106490963093023847315560025\ 477169376164365042536688559994140155058094946223202957075832\ 611993276199866424759164594817727294720307633597480289711815\ 496991872281261569527767888721253039470089461149792442700000n + (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.