½ = 1/x² + 1/y² +...+ 1/z²

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

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