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 Get one half by squaring (Posted on 2004-09-20)
Can you solve the following equation?

½ = 1/x² + 1/y² +...+ 1/z²

All variables must be different, positive integers, and there must be a finite number of terms.

 See The Solution Submitted by Federico Kereki Rating: 4.2500 (12 votes)

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 The best that my program can do | Comment 14 of 31 |

A computer program comes up with the following variables:

2,3,4,5,6,11,54,519,59429

which totals about 0.4999999999999980848

and has some indication that starting an infinite series at 22686937 would help complete the total.

The program follows, and actually assumes that 2 will be one of the variables and starts from there without reporting the 2.

DECLARE SUB try (n#)
CLEAR , , 4000

DEFDBL A-Z
DIM SHARED lim, tot, used, remain, trm(50), howMany

pi = ATN(1) * 4
lim = pi * pi / 6 - 1.25
tot = 0
used = 0
remain = lim

try 3

SUB try (n)
term = 1 / (n * n)
rSave = remain
remain = remain - term
IF ABS(term + used - .25) < 1E-12 THEN
FOR i = 1 TO howMany
PRINT trm(i)
NEXT
PRINT n
END IF
IF term + used <= .25 + 1E-12 THEN
uSave = used
used = used + term
howMany = howMany + 1
trm(howMany) = n
try n + 1
howMany = howMany - 1
used = uSave
END IF
IF remain >= .25 - used THEN
rSave = remain
uSave = used
need = 1 / SQR(.25 - used)
new = -INT(-need)
FOR i = n + 1 TO new - 1
term = 1 / (i * i)
remain = remain - term
NEXT i
try new
used = uSave
remain = rSave
END IF
remain = rSave
END SUB

with output

3
4
5
6
11
54
519
59429
22686937
22686938
22686939
22686940
22686941
22686942
22686943
22686944
22686945
22686946
22686947
22686948
22686949
22686950
22686951
22686952
22686953
22686954
22686955
22686956
22686957
22686958
22686959
22686960
22686961
22686962
22686963

 Posted by Charlie on 2004-09-21 10:29:01

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