Problems & Puzzles: Puzzles

 Puzzle 364. P2 = Q*R J. S. Madachy, in his nice "Madachy's Mathematical Recreations", on pp. 159 writes: (493827156)2 = (246913578)*(987654312) Are there other such results? Madachy is asking for solutions to  P2 = Q*R where P, Q & R are zero-less pandigitals. I have found that the example given by Madachy correspond to a model which accepts other 605 solutions, the smallest one being: 2469135782 = 123456789*493827156 and the largest being the reported by Madachy. *** Out of the model used by Madachy, I found three more solutions: 2315976842 = Q*R 3215967842 = Q*R 4826931572 = Q*R Questions. 1. What are the Q & R values for the 3 solutions given above? 2. Are there other models/solutions out of the 608 mentioned above? 3. Redo the exercise for the 10-digits pandigitals.

Contributions came from Antoine Verroken & Farideh Firoozbakht:

***

Antoine wrote:

231597684^2 = 325196748 * 164938572
321596784^2 = 214397856 * 482395176
482693157^2 = 398746521 * 584312769

***

Farideh wrote:

Number of solutions for the equation  P^2 = (P/2)*(2P)  where P, P/2 and 2P

are 9-digits pandigitals is 608 (so I think the number 605 isn't correct).
As you wrote 246913578 & 493827156 are smallest and largest of them.

Except the model  P^2 = (P/2)*(2P)  which found by Madachy and the three
following models,

1. P^2 = (P/(271/193))*((271/193)P)       ( 231597684^2 = 164938572*325196748 )
2. P^2 = (P/(3/2))*((3/2)P)                     ( 321596784^2 = 214397856*482395176 )
3. P^2 = (P/(23/19))*((23/19)P)              ( 482693157^2 = 398746521*584312769 )

that you have found there exists only 11 other models which one of them has two solutions and each of the others has only one solutions.

1. P^2 = (P/(7/4))*((7/4)P)                     P = 279516384 & P = 483615972

2. P^2 = (P/(23/14))*((23/14)P)                     P = 254189376
3. P^2 = (P/(109/91))*((109/91)P)                 P = 327981654
4. P^2 = (P/(27/13))*((27/13)P)                     P = 352714986
5. P^2 = (P/(21/11))*((21/11)P)                     P = 357486129
6. P^2 = (P/(6521/5022))*((6521/5022)P)       P = 589472316
7. P^2 = (P/(9091/7813))*((9091/7813)P)       P = 639251847
8. P^2 = (P/(7/6))*((7/6)P)                            P = 769321854

So number of all the solutions is  608+(1+1+1)+ 2+(1+1+1+1+1+1+1) = 620.

*****************

The exercise for the 10-digits pandigitals:

Models which more  than              Number of    The smallest      The largest
one  solution                           solutions       solution            solution

1. P^2 = (P/2)*(2P)                               6480    2046913578       4938271560
2. P^2 = (P/(3/2))*((3/2)P)                      19       2019763458       4928537610
3. P^2 = (P/(5/2))*((5/2)P)                       8        2685973140       3746912580
4. P^2 = (P/(5/4))*((5/4)P)                       4        2469135780       6295371480
5. P^2 = (P/(7/4))*((7/4)P)                       3        2795163840       4836159720
6. P^2 = (P/(4/3))*((4/3)P)                       3        1385967024       5213904768
7. P^2 = (P/(7/6))*((7/6)P)                       2        3014596872       7693218540
8. P^2 = (P/(31/27))*((31/27)P)                2        5293876014       5704236189
9. P^2 = (P/3)*(3P)                                 2       3071246895       3104692875

Also we have the following nice equation.

5318402976^2=3071952864*9207631584=4603815792*6143905728

So P=5318402976 is a solution for the both equations
P^2=(P/(393/227))*(P*(393/227))   &  P^2=(P/(454/393))*(P*(454/393))

All other such equations are:

2469135780^2 = 1234567890*4938271560 = 1975308624*3086419725
2685973140^2 = 1074389256*6714932850 = 5371946280*1342986570
3214768590^2 = 1285907436*8036921475 = 1607384295*6429537180
3569841270^2 = 1427936508*8924603175 = 1784920635*7139682540
3681274590^2 = 1472509836*9203186475 = 1840637295*7362549180
3681472590^2 = 1472589036*9203681475 = 1840736295*7362945180
3714926580^2 = 1485970632*9287316450 = 1857463290*7429853160
3726914580^2 = 1490765832*9317286450 = 1863457290*7453829160
3746912580^2 = 1498765032*9367281450 = 1873456290*7493825160
4938271560^2 = 2469135780*9876543120 = 3950617248*6172839450

So the set of 8 solutions of the third models is a subset of the set of solutions
of the first model. And two members of the set of 4 solutions of the fourth model
are two solutions of the 6480 solutions for the first model.

There exists 96 other models where each of them has only one solution.
These models are P^2 = (P/r)*(r*P) where r is a member of the following set.

{32/31, 163/157, 4036/3877, 6776/6495, 24/23, 106/101, 212/201,
21812/20537, 17/16,14/13, 257/235, 906/817,381/343,19/17,137/122,
26024/23171,1188/1051, 74/65, 49/43, 8/7, 454/393, 15033/13012,
13009/11248, 9091/7813, 316/271, 109/91, 31565/26151, 23/19,
678/557, 9947/8069, 123/98,1913/1522,207/164,13307/10516,
71/56, 2854/2251, 65/51,9/7, 378/293, 6521/5022,2069/1566, 41/31,
12058/9091, 12716/9463, 658/489, 31/23, 5813/4264,1606/1175,
525/383, 6613/4812,62/45, 271/193, 9091/6451, 32234/22035, 20622/13999,
28/19,34/23,114/77, 489/329, 8069/5428, 1306/855, 25394/16451,14/9,
29213/18527, 33022/20893, 23/14, 4918/2959, 5/3,28601/16902, 135/79,
393/227, 37/21, 2053/1144, 4274/2359, 297/163, 3627/1936, 206/109,
21/11, 5481/2815, 29785/15189, 23811/11951,291/146, 30535/15208,
29693/14634, 26787/13049, 27/13, 25253/11919, 2682/1261, 3339/1525,
11/5,4533/2006,34/15, 245/108,19909/8647, 58/23, 9947/3908}

Farideh added the following new issues:

I think solutions for the following five equations where  P, Q, R & Z are distinct
9-digits (10-digits)  pandigitals and dot means concatenations are interesting too.

P^2 = Q.R          P.P = Q*R

P^3 = Q*R*Z    P^3 = Q.R.Z     P.P.P = Q*R*Z

And the examples are:

The two solutions 4253907186^2 = 1809572634.7102438596 &
6432015987^2 = 4137082965.7023584169  are all of the ten-digits pandigital
solutions for the equation P^2=Q.R and there is no 9-digits solutions.

The equation  P^3 = Q.R.Z hasn't any pandigital solutions.
The equation P.P = Q*R where P, Q & R are 10-digits pandigitals has only 7 solutions. 1376925480.1376925480 = 2658703194*5178936420 is the smallest of them and 5041893276.5041893276 = 6590381724*7650381249 is the largest. There is no nine-digits solution for the equation P.P = Q*R.

***

Giovanni Resta wrote (March 2011):

At the end of the solutions for puzzle 364,
Farideh proposed some extensions for your puzzle,
and provided some of the solutions.

These are the solutions for the subproblems which
were still open.

Subproblem P^3 = Q*R*Z
-------------
1) Zero-less pandigital:
There are 3 solutions:
284397156^3 = 162598374 * 186537249 * 758392416
375192468^3 = 187596234 * 354218976 * 794815623
378691254^3 = 189345627 * 531647982 * 539481276

2) Pandigital:
There are 44 solutions, from:
1742598036^3 = 1306948527 * 1548976032 * 2613897054 to
7068953124^3 = 4917532608 * 7935068241 * 9052476183.

There is a multiple solution in which the 6 numbers involved
are all distinct:

3587964120^3 =  1307489625 * 5203147968 * 6789532104
=  2539174608 * 3802461975 * 4783952160.

Subproblem P.P.P=Q*R*Z
--------------
1) Zero-less pandigital:
No solutions since every number of the form P.P.P
is divisible by 1+10^9+10^18 which among its
prime factors has 440334654777631 which is greater
than any zero-less pandigital.

2) Pandigital:
No solutions since every number of the form P.P.P
is divisible by 1+10^10+10^20 which among its
prime factors has 2906161 and no pandigital number
is divisible by that prime.

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