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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.4 ninedigital = quadruple solutions for A2 + B2

 A and B restricted to 4-digit numbers 
No luck, there are no strictly quadruple 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.
143869725 = 99152 + 67502
= 95252 + 72902
= 87662 + 81872
= 86612 + 82982
[ eight more when unrestricted ]

 A and B restricted to 5-digit numbers 
415869273 = 110372 + 171482
= 121922 + 163472
= 132722 + 154832
= 143072 + 145322
                     [ 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
= 110852 + 149702
= 121622 + 141092
= 126752 + 136502
= 131462 + 131972
[ smallest ]





[ 13 more when unrestricted ]

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

 A and B restricted to 4-digit numbers 
No 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
= 109652 + 173102
= 121622 + 164912
= 124262 + 162932
= 130352 + 158102
= 143582 + 146192
                     [ smallest ]






[ 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
= 56492 + 96422
= 60542 + 93932
= 63422 + 92012
= 65462 + 90572
= 71672 + 85742
= 73772 + 83942
[ smallest ]







[ 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
1052736489 = 9923 + 87492   Both basenumbers are indivisible by either 3 or 9
8197604352 = 20162 + 20163   See you next year !
 Case P.P.4 ninedigital = double solutions for A3 + B2
1023597648
= 9993 + 51572
= 9723 + 102602

Database   

Note: this list is very lengthy !
Creating a complete list might be unrealistic, so I will replace it
with a UBASIC program someday.


A000193 Prime Curios! Prime Puzzle
Wikipedia 193 Le nombre 193

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