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

1233 = 12² + 33²

990100 = 990² + 100²

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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 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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