I'm thinking of a number, x, whose last 3 digits are somewhere between 400 and 500.
You could probably guess from those digits that it might be a square or a cube, but in fact it turns out to be both, and in fact the smallest possible such number.
What are the prime factors of x?
Extra credit: feeling adventurous, I next computed the smallest number, y, having the same last 3 digits, that was a seventh power, as well as a square and a cube.
What are the prime factors of y?
The program listing below finds that
3^12 [531441] is both a square
[3^6] and a cube
[3^4] – PART 1.
Although the program uses “double” it still would not have been able to evaluate the end result had “long” been defined as the highest value available is 1.7E+308.
To achieve the desired end the program works with truncated values repeating these until a 7th power is achieved which meets the demands of being a square, a cube and terminating in 1441.
One can achieve a lengthier table similar to PART 2 with Excel:
ColA
Row1 =IF(INT(B1/7)=B1/7,INT(B1/7),””)
ColB ColC ColD ColE
12 1441 531441 =RIGHT(D1,4)
Row2 ColA
[copy formula in A1]
ColB ColC ColD ColE
= B1+12 1441 =C1+C2 [copy formula in E1]
Copying contents of Row2 down to Row126 yields:
216 1512 1441 1441 1441.Without further comment on it I‘ll leave the reader to contemplate that table.
1512 = 7 * 216 1512 = 2 * 756 and 1512 = 3 * 504 so we have:
7 * (3^216), 2 * (3^756) and 3* (3^504) which all equal 3^1512.
The highest value Excel will handle here is 3^646 = 1.6609E+308.
3^1512 = 3^600 * 3^600 * 3^312
or = (1.8739E+286)* (1.8739E+286)*(7.275E+148)
= 2.5545E+721
PART 1* 729 s 531441 cr 81
PART 2 12 84 1441 5681 5681
24 168 1441 3761 3761
36 252 1441 6241 6241
48 336 1441 5121 5121
60 420 1441 2401 2401
72 504 1441 81 0081
84 588 1441 161 0161
96 672 1441 4641 4641
108 756 1441 5521 5521
120 840 1441 4801 4801
132 924 1441 4481 4481
144 1008 1441 6561 6561
156 1092 1441 3041 3041
168 1176 1441 5921 5921
180 1260 1441 7201 7201
192 1344 1441 8881 8881
204 1428 1441 2961 2961
216 1512 1441 1441 1441OPEN "c:\qb64\work\br_cube.txt" FOR OUTPUT AS #1
DEFDBL A-Z
n = 2: done = 0
DO
LOCATE 1, 50: PRINT n
s = n * n
PRINT n, s ^ (1 / 3), INT(s ^ (1 / 3))
cr = INT(s ^ (1 / 3) + .5)
IF s = cr * cr * cr THEN
crval$ = RIGHT$(STR$(s), 3)
IF VAL(crval$) > 400 AND VAL(crval$) < 500 THEN
' PRINT s ^ (1 / 2), s ^ (1 / 3)
PRINT "*"; n, "s"; s, "cr"; s ^ (1 / 3)
PRINT
PRINT #1, "Part 1"
PRINT #1, "*"; n, "s"; s, "cr"; s ^ (1 / 3)
PRINT #1,
PRINT #1,
done = 1
END IF
END IF
n = n + 1
LOOP WHILE done <> 1
done = 0
REM PART 2
y = 1
tm = 1441
done = 0
n = 0
PRINT #1, "PART 2"
DO
n = n + 12
y = y * tm
term$ = RIGHT$(STR$(y), 4)
y = VAL(RIGHT$(STR$(y), 4))
IF INT(n / 7) = n / 7 THEN
PRINT n / 7; n; 1441; y; term$
PRINT #1, n / 7; n; 1441; y; term$
END IF
IF INT(n / 7) = n / 7 AND RIGHT$(term$, 3) = "441" THEN
done = 1
END IF
'' DO
''LOOP WHILE INKEY$ <> CHR$(32)
LOOP WHILE done <> 1
CLOSE 1
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Posted by brianjn
on 2013-01-30 17:37:24 |
Programming and Excel
