WON TOPIC 193

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[ February 1, 2015 ]
Exploring nine- and pandigitals
expressible as sums of two squares
B.S. Rangaswamy (email)

B.S. Rangaswamy enclosed me 2_tier constituents of powers for ninedigitals and pandigitals.
Some have double, triple, quadruple, quintuple and even a sevenfold set of constituents.
He used clever Excel table programming to get some selected (restricted) results
where one of the two powers is an eight-digit number in the case of ninedigitals
and a nine-digit number in the case of pandigitals (the 2nd powers are only resultants).
The filtering and sorting was speeded up with the help of his grandson Puneeth
and daughter in law Indira Ramesh.
Alas, I cannot reproduce the programs in HTML, so I will stick to
displaying partly results and some magical curios for some chosen categories
but without the restrictions imposed by B.S. Rangaswamy.
Interested and involving readers may try to discover existence of other
strange constituents for any of the remaining vast number of nine- and/or pandigitals.

Ninedigital highlights

Here are already some curios (multi solutions per ninedigital allowed)
143625897 = 9876² + 6789² Note the reversals !

143752968 = 8478² + 8478²   Twice the same term !
145897362 = 8541² + 8541²   Here also twice the same term !
162973458 = 9027² + 9027²   And again twice the same term !

143689725 = 8850² + 8085²   Both terms are anagrams of each other !
145973682 = 9861² + 6981²   Bis - both terms are anagrams of each other !
149683725 = 8754² + 8547²   Ter - both terms are anagrams of each other !

Statistical information and more curios are to be found
in one of my dedicated ninedigits pages. Please visit also
Statistics and curios

You noticed already that all values A and B are multiples of 3. How come ?

This due to the fact that every ninedigital is divisible by 9 (all digits sums to 45).
In order to be able to separate the 9-factor from the sum of squares, each
value A and B must be a multiple of 3. Idem dito for the pandigitals.
N = A² + B²
9 * N' = 9 * ( A'² + B'² )
9 * N' = ( 9 * A'² ) + ( 9 * B'² )
9 * N' = ( 3 * A' )² + ( 3 * B' )²

Case N.1 ninedigital = strictly single solutions for A² + B²

A and B restricted to 4-digit numbers

123456978 = 5223² + 9807²	[ Smallest ]
123458976 = 7476² + 8220²
Many more solutions exist	[ Solutions must be < 200000000 ]
193675842 = 9729² + 9951²
194265378 = 9747² + 9963²	[ Largest ]

A and B restricted to 5-digit numbers

213576498 = 10137² + 10527²
312457986 = 12069² + 12915²
412536978 = 12747² + 15813²
512374986 = 13815² + 17931²
612345978 = 10803² + 22263²
712345896 = 14214² + 22590²
812356794 = 19365² + 20913²
912346857 = 13101² + 27216²	[ A selection of solutions ]

Case N.2 ninedigital = strictly double solutions for A² + B²

A and B restricted to 4-digit numbers

Solutions must be < 200000000.

123475986 = 6825² + 8769² = 6969² + 8655²
152367489 = 9567² + 7800² = 9540² + 7833²	[ Many more solutions exist ]

A and B restricted to 5-digit numbers

245193768 = 10038² + 12018² = 10842² + 11298²
312685749 = 10695² + 14082² = 11310² + 13593²
412365978 = 11133² + 16983² = 14013² + 14697²
512387493 = 13686² + 18030² = 14586² + 17310²	[ Many more solutions exist ]

Case N.3 ninedigital = strictly triple solutions for A² + B²
Note that these triples are sparingly scattered.

A and B restricted to 4-digit numbers

135697428 = 6402² + 9732²
= 6828² + 9438²
= 8052² + 8418²

[ a strictly _unique_ triple solution ]

A and B restricted to 5-digit numbers

326987541 = 10455² + 14754²
= 11946² + 13575²
= 12246² + 13305²

[ Smallest ]

More solutions exist.

Case N.5 ninedigital = quintuple solutions for A² + B²

A and B restricted to 4-digit numbers

There exist no strictly quintuple solutions for A² + B²

The next best thing is to search for them as a subset in higher x-fold solutions,
as for instance in the following 12-fold example from B.S. Rangaswamy.

143697825 = 9777² + 6936²
= 9732² + 6999²
= 9444² + 7383²
= 9276² + 7593²
= 8913² + 8016²

[ seven more when unrestricted ]

A and B restricted to 5-digit numbers

I haven't encountered a strictly quintuple solution of this kind. It doesn't exist.

The next best thing is to search for them as a subset in higher x-fold solutions,
as for instance in the following 18-fold example.

346978125 = 10110² + 15645²	[ smallest ]
= 11085² + 14970²
= 12162² + 14109²
= 12675² + 13650²
= 13146² + 13197²	[ 13 more when unrestricted ]

Case N.6 ninedigital = sextuple solutions for A² + B²

A and B restricted to 4-digit numbers

o luck, there are no strictly sextuple solutions for A² + B²

The next best thing is to search for them as a subset in higher x-fold solutions,
but alas, here also there are NO solutions.

A and B restricted to 5-digit numbers

No luck either, there are no strictly sextuple solutions for A² + B²

The next best thing is to search for them as a subset in higher x-fold solutions,
as for instance in the following 12-fold example.

419867325 = 10014² + 17877²	[ smallest ]
= 10965² + 17310²
= 12162² + 16491²
= 12426² + 16293²
= 13035² + 15810²
= 14358² + 14619²	[ six more when unrestricted ]
Other subset sextuple solutions are 469371825, 469738125, 487392165, 569384712, 597632841, 618394725, 631849725, 634189725, 637184925, 643198725, 679238145, 679431285, 679843125, 684793125, 684925137, 689173425, 691437825, 692731845, 697354281, 698317425, 718643925, 724961385, 736841925, 741268593, 761943825, 764318925, 764893125, 769413285, 769438125, 781936425, 783952416, 784319562, 793461825, 794318625, 813764925, 813796425, 817923465, 819347625, 825641973, 841796325, 845629317, 847693125, 861473925, 864173925, 874169325, 876941325, 893647521, 893765412, 894132765, 894673125, 895734216, 896473125, 912374865, 918376425, 925764138, 926457381, 932147658, 932765418, 934768125, 936784125, 938671425, 942716385, 943218765, 946783125, 947138625, 964873125, 967354128, 967843125, 971438625, 974683125, 976412385, 976823145, 981324765, 986371425 and 987342165

Case N.7 ninedigital = sevenfold solutions for A² + B²

A and B restricted to 4-digit numbers

There exist no strictly 7-fold solutions for A² + B²

The next best thing is to search for them as a subset in higher x-fold solutions,
as for instance in the following 24-fold example from B.S.Rangaswamy.

139876425 = 9876² + 6507²
= 9699² + 6768²
= 9612² + 6891²
= 9213² + 7416²
= 9045² + 7620²
= 9012² + 7659²
= 8445² + 8280²	[ 17 more when unrestricted ]

But the first occurence happens within the following 16-fold ninedigital

124879365 = 5439² + 9762²	[ smallest ]
= 5649² + 9642²
= 6054² + 9393²
= 6342² + 9201²
= 6546² + 9057²
= 7167² + 8574²
= 7377² + 8394²	[ nine more when unrestricted ]

A and B restricted to 5-digit numbers

There exist no strictly 7-fold solutions for A² + B²

The next best thing is to search for them as a subset in higher x-fold solutions,
as for instance in the following 8-fold example.

627498153 = 11277² + 22368²
= 12732² + 21573²
= 13203² + 21288²
= 15483² + 19692²
= 15813² + 19428²
= 15912² + 19347²
= 17157² + 18252²	[ one more when unrestricted ]
Other subset sevenfold solutions are 631984725, 681257394, 683791425, 719486325, 734891625, 738469125, 738694125, 745631289, 746193825, 748931625, 764391825, 781249365, 781634925, 789245613, 791648325, 792813645, 813469725, 817364925, 831476925, 832176945, 841396725, 846139725, 846912573, 849273165, 851327649, 861734925, 861934725, 864379125, 864713925, 867495213, 867913425, 876931425, 879361425, 893417265, 896714325, 914732685, 914863725, 916873425, 917483265, 918476325, 924163578, 938647125, 946731825, 961438725, 964378125, 978146325, 982675413, 983167425, 984127365 and 986731425

Case N.8 ninedigital = eightfold solutions for A² + B²

A and B restricted to 5-digit numbers

895723146 = 12015² + 27411² = 15789² + 25425² = 16761² + 24795² = 17205² + 24489² = 17739² + 24105² = 20115² + 22161² = 20511² + 21795² = 20985² + 21339²	[ a strictly _unique_ eightfold solution ]
Other subset eightfold solutions are 463879125, 647193825, 691847325, 719348625, 761438925, 763849125, 789346125, 794386125, 794631825, 817639425, 817693425, 821749365, 831249765, 846731925, 879461325, 879463125, 971382465 and 982715634

Case N.9 ninedigital = higher-fold solutions for A² + B²

A and B restricted to 5-digit numbers

9-fold solutions	728931645 subset from a 16-fold 734189625 subset from a 16-fold 739241685 subset from a 16-fold 837612945 subset from a 16-fold 873469125 subset from a 16-fold 893127645 subset from a 16-fold 897134625 subset from a 16-fold 912634785 subset from a 16-fold 914273685 subset from a 16-fold 948376125 subset from a 16-fold 963417285 subset from a 16-fold 971436258 subset from a 16-fold 981342765 subset from a 16-fold
10-fold solutions	417938625 subset from a 24-fold 783961425 subset from a 18-fold 987634125 subset from a 16-fold
11-fold solutions	439817625 subset from a 32-fold 763498125 subset from a 20-fold 796843125 subset from a 20-fold 946837125 subset from a 16-fold 961342785 subset from a 16-fold 971238645 subset from a 16-fold 973864125 subset from a 16-fold
12-fold solutions	698341725 subset from a 24-fold 726193845 subset from a 24-fold 968417325 subset from a 18-fold
13-fold solutions	839147625 subset from a 24-fold 913276845 subset from a 24-fold
14-fold solutions	869371425 subset from a 24-fold
15-fold solutions	943678125 subset from a 24-fold

Why is it that some ninedigitals have more 'sum of squares' solutions than others?

This is due to the fact every ninedigital has its unique factorization.
Take for instance 439817625 which has nine small prime factors.
3 * 3 * 5 * 5 * 5 * 13 * 17 * 29 * 61
This allows for many combinations and thus eligible solutions whereas e.g.
123458679 = 3 * 3 * 13717631
has only three prime factors and hence impossible to express as a sum of squares.

Case N.N.1 Lowest and highest ninedigitals for A² + B²

A and B restricted to 4-digit numbers

123456978 = 5223² + 9807²   is the lowest ( from a strictly single solution )

194265378 = 9747² + 9963²   is the highest ( in case of a strictly single solution )
198463725 = 9942² + 9981²   is one highest version from a multi_3 solution

A and B restricted to 5-digit numbers

213458697 = 10059² + 10596²   is the lowest ( ie. the highest version from a multi_2 solution )
213576498 = 10137² + 10527²   is the lowest ( in case of a strictly single solution )

987641253 = 11823² + 29118²   is the highest ( in case of a strictly single solution )
987654321 = 18264² + 25575²   is one highest version from a multi_3 solution

Case N.N.2 Apart from squares there are cubes & other powers
as 2_tier constituents of ninedigitals
but that will be a topic for a separate investigation in the future.

125734689 = 450³ + 5883²
136725489 = 8433² + 90⁴
137865924 = 432³ + 7566²
137965248 = 447³ + 6975²
138267594 = 417³ + 8109²
138427956 = 435³ + 7491²
138629457 = 8841² + 6¹⁰
139487652 = 96⁴ + 7386²
139842756 = 96⁴ + 7410²
139854276 = 8910² + 6¹⁰
142687953 = 438³ + 7659²

Case N.N.3 ninedigital = single solutions for A³ + B²

125734689 = 450³ + 5883²
145273689 = 462³ + 6831²

Case N.11 ninedigital = single solutions for A⁴ + B²

123497685 = 93⁴ + 6978²
123498756 = 96⁴ + 6210²

B.S. Rangaswamy wishes all of us a happy new year 2015 ( = 11³ + 26² + 2³)

Pandigital highlights

Case P.1 pandigital = single solutions for A² + B²
( A and B not restricted ).

Lots of solutions are known...

1029846537 = 30096² + 11139²

2345679018 = 29283² + 38577²

6098752314 = 37767² + 68355²

Case P.2 pandigital = double solutions for A² + B²
( A and B not restricted ).

Many more solutions exist...

1023457698	= 31533² + 5397² = 30363² + 10077²

1029853476	= 31530² + 5976² = 30840² + 8874²

9712304685	= 31554² + 93363² = 55758² + 81261²

Case P.3 pandigital = triple solutions for A² + B².

A and B restricted to 5-digit numbers

1034962857

= 10221² + 30504²
= 16059² + 27876²
= 21261² + 24144²

Case P.4 pandigital = quadruple solutions for A² + B²
( A and B not restricted ).

1023698754	= 31773² + 3765² = 31515² + 5523² = 31215² + 7023² = 30777² + 8745²

1083456729	= 31005² + 11052² = 30627² + 12060² = 30420² + 12573² = 29475² + 14652²

Case P.5 pandigital = quintuple solutions for A² + B²

Who can provide me with the first example ?

Case P.6 pandigital = sextuple solutions for A² + B²

Who can provide me with the first example ?

Case P.7 pandigital = sevenfold solutions for A² + B²

Who can provide me with the first example ?

Case P.P.1 Lowest and highest pandigital for A² + B²
( A and B not restricted ).

1023456789 = 31383² + 6210² is the lowest version 1
1023456789 = 29367² + 12690² is the lowest version 2

9876435210 = 30603² + 94551² is one of eight highest versions

Case P.P.2 Already one nice curio combining two numbertypes

(pandigital) 1024397685 = 31482² + 5769² (ninedigital)

Case P.P.3 More nice curios in case of higher exponents