Suppose a and b are positive integers. We all know that aČ+2ab+bČ is a perfect square. Give an example where also aČ+ab+bČ is a perfect square. How many such examples exist?
(In reply to
re: Partial Solution by Charlie)
Here's just the beginning of what seems a never-ending list of the beginnings of families of solutions. The only member of the family shown is the one in which the gcd of the two numbers is 1.
3 5
7 8
5 16
11 24
7 33
13 35
16 39
9 56
32 45
17 63
40 51
11 85
19 80
55 57
40 77
24 95
13 120
23 120
65 88
69 91
25 143
56 115
15 161
75 112
32 175
104 105
56 165
29 195
17 208
87 160
85 168
31 224
119 145
72 203
19 261
93 187
64 221
144 155
95 217
40 279
10 T=3
20 while (Ct<40)
30 for I=1 to int(T/2)
40 J=T-I
50 if gcd(I,J)=1 then
60 :V=I*I+I*J+J*J
70 :Sr=int(sqrt(V)+0.5)
80 :if Sr*Sr=V then print I,J:Ct=Ct+1:endif
90 next
95 T=T+1
100 wend
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Posted by Charlie
on 2006-04-25 10:05:05 |
re(2): Partial Solution
