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8 = 4 + 4 (Posted on 2004-08-26) Difficulty: 3 of 5
In the line of

1233 = 12 + 33

990100 = 990 + 100

can you find an eight digit number with the same property? Even longer numbers?

See The Solution Submitted by Federico Kereki    
Rating: 3.0000 (4 votes)

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Solution re: more values -- program | Comment 8 of 11 |
(In reply to more values by Charles Lutes)

If 2n is the number of digits in the total then

(10^n)*a + b = a^2 + b^2

and

a^2 - (10^n)*a + b^2 - b = 0

and

a = (10^n +/- sqrt(10^(2*n) - 4*(b^2 - b)) / 2

UBASIC allows for extra precision variables, so:

    5   N=10^1
    6   repeat
   10   repeat
   30    D=(N^2-4*(B*B-B))
   40    Sd=int(sqrt(D)+0.5)
   50    if Sd*Sd=D then
   60     :print B,(N-Sd)//2,(N+Sd)//2
   70    B+=1
   80   until 4*(B*B-B)>=N^2
   90   N*=10
  100   until N>10^9

producing:

 0               0               10
 1       0       10
 33      12      88
 100     10      990
 2353    588     9412
 9901    990     99010
 38125   17650   82350
 43776   25840   74160
 321168          116788   883212
 328768          123288   876712
 773101          60130   9939870
 990100          99010   9900990
 2663025         768180          9231820
 2846976         889680          9110320
 3448276         1379310         8620690
 3604525         1534830         8465170
 10243728        1060588         98939412
 12888513        1689688         98310312
 15600625        2496100         97503900
 17428225        3135760         96864240
 25002048        6699912         93300088
 27318513        8122812         91877188
 31180401        10913140        89086860
 38526913        18130312        81869688
 40202128        20271412        79728588
 42355776        23429560        76570440
 45440001        29138400        70861600
 46547313        31742188        68257812
 47470848        34299088        65700912
 48531600        37971540        62028460
 49680625        44357700        55642300
... (program manually terminated)

Ignoring the first two lines, examining the next five:

 33              12              88
 100     10      990
 2353    588     9412
 9901    990     99010
 38125   17650   82350

The ending digits are shown on the left and the closing digits on the right. So 1233 and 8833 are shown on the first (actually third) line.  Then, for b=100, we show not only the 990100, but also 010100 = 010^2 + 100^2, which is unsatisfactory due to the leading zero, as is the case for b=9901-- with both shown prefixes.  But whenever the number of digits in the middle column equals the number of digits in the left and right hand columns, there are two results for the given b, as in 1765038125 and 8235038125.  ... and so on down the rest of the list, to the largest I've shown: 44357700^2+ 49680625^2 = 4435770049680625 and 55642300^2 +  49680625^2 =             5564230049680625 .

 

Edited on August 27, 2004, 11:42 am
  Posted by Charlie on 2004-08-27 11:35:24

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