#1 10. 12. 2013 05:03
- stuart clark
- Příspěvky: 1015
- Reputace: 7
integer ordered pairs
Total no. of positive integer ordered pairs 
in 
Total no. of positive integer ordered pairs in 
If 
, Then no. of positive integer ordered pairs 
is
Total no. of positive integer ordered pairs in 
in 
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#2 11. 12. 2013 17:07
Re: integer ordered pairs
. 
cases:
, then 
, so 
, so 
. Then we've got 
, so 
. 
and 
or 
and 
or 
and 
or 
, 
, no solution. 
, then 
, so 
, so 
. This case we've already solved. 
, then 
, so 
, so 
. We've got 
, or 
.
we got solution only for 
, then 
. Suppose 
. 
, then 
, so 
and 
, so 
, contradiction. So 
. 
, then 
. For 
or 
it's not true. We prove 
for 
. For 
it is true. We need to prove
, which is equivalent to 
, which is true for 
, then 
. For 
for 
. For 
, which is equivalent to 
, which is true for 
, then 
or 
, no solution for 
.
, so 
, so 
, so 
and 
. 
(then no solution) or 
(then 
, so no solution). 
, so 
or 
. 
, impossible. 
. For 
no solution. Let 
. If 
, which is impossible. So 
. Solution only for 
. 
.
, so 
. For 
, which is impossible. So 
, then 
, which is impossible. So 
, so again we must have 
. We've got solution only for 
. Then 
.
. So 
. If 
, which is impossible. So 
. We've got solution only for 
or 
.
solutions.1^6 - 2^6 + 3^6 = 666
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