Puzzle 364. P2 = Q*R

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

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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.

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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.

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