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 Make it solvable (Posted on 2005-05-05)
a, b, and x are positive integers such that

sqrt(a) + sqrt(b) = sqrt(x)

How many possible values of x less than or equal to 1000 are there?

 See The Solution Submitted by Jer Rating: 3.6667 (3 votes)

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 re(4): First thoughts... ...(spoiler?) | Comment 6 of 16 |
(In reply to re(3): First thoughts... ...(spoiler?) by Charlie)

The program I used to count solutions was:

DEFDBL A-Z
OPEN "makesolv.txt" FOR OUTPUT AS #2
OPEN "makesol2.txt" FOR OUTPUT AS #3
FOR x = 1 TO 1000
sx = SQR(x)
a = 1
flag = 0: psqflag = 0
DO
sa = SQR(a)
psq = 0
isa = INT(sa + .5)
IF isa * isa = a THEN psq = 1
sb = sx - sa
b = sb * sb
IF b < a THEN EXIT DO
rb = INT(b + .5)
IF ABS(b - rb) < ABS(b) / 1E+13 THEN
PRINT USING "#### ####.####### ####"; a; b; x;
PRINT #2, USING "#### ####.####### ####"; a; b; x;
ct2 = ct2 + 1
flag = 1
IF psq THEN
psqflag = 1
PRINT " *"
PRINT #2, " *"
psqct2 = psqct2 + 1
IF x <> pxpsq THEN
PRINT #3,
pxpsq = x
END IF
PRINT #3, USING "#### #### ####"; a; b; x
ELSE
PRINT
PRINT #2,
END IF
END IF
a = a + 1
LOOP
IF flag THEN
ct = ct + 1
IF psqflag THEN ctpsq = ctpsq + 1
PRINT
PRINT #2,
END IF
NEXT x
PRINT ct, ct2, ctpsq, psqct2

where the final counts came as:

355           904           30            240

indicating 355 values of x through 904 different sums, where 30 values of x were obtained through perfect squares as the integers, among 240 such sums involving perfect squares (that is sums of integers).

 Posted by Charlie on 2005-05-05 19:55:13

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