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A260825
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List of pairs of positive integers [b,c], in increasing order of b*c, whose sum and product are both squares.
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1
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2, 2, 8, 8, 5, 20, 9, 16, 2, 98, 18, 18, 10, 90, 32, 32, 20, 80, 36, 64, 50, 50, 8, 392, 25, 144, 17, 272, 72, 72, 52, 117, 2, 3362, 45, 180, 98, 98, 81, 144, 20, 605, 64, 225, 40, 360, 18, 882, 128, 128, 26, 650, 80, 320, 162, 162, 49, 576, 5, 7220, 144, 256, 200, 200, 37, 1332, 32, 1568, 13, 4212, 98, 578
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OFFSET
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1,1
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COMMENTS
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Pairs have been put in increasing order of b*c, thus [9, 16] comes after [2, 98] (since 9*16 < 2*98).
b and c are listed with b <= c.
First b*c that has more than one pair is 120^2 corresponding to [b,c] = [64, 225] and [40, 360].
For a given b*c, pairs are placed in increasing order of c. (End)
The squares (b*c) are 4, 64, 100, 144, 196, 324, 900, 1024, 1600, 2304, 2500, 3136, 3600, 4624, 5184, ..., .
The square roots of (b*c) are 2, 8, 10, 12, 14, 18, 30, 32, 40, 48, 50, 56, 60, 68, 72, 78, 82, 90, 98, ..., .
The sums (b+c) are 4, 16, 25, 25, 100, 36, 100, 64, 100, 100, 100, 400, 169, 289, 144, 169, 3364, 225, ..., . (End)
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LINKS
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EXAMPLE
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[5,20] is such a pair, since 5 + 20 = 25 = 5^2 and 5*20 = 100 = 10^2.
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MAPLE
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count:= 0: Res:= NULL;
for t from 1 while count < 200 do
tres:= NULL;
Q:= select(q -> q^2 <= 4*t^2, numtheory:-divisors(4*t^2));
nt:= 0:
for q in Q do
r:= 4*t^2/q;
if (r-q) mod 2 <> 0 then next fi;
s2:= (q+r)/2;
if not issqr(s2) then next fi;
s:= sqrt(s2);
d:= (r-q)/2;
b:= (s2+d)/2;
c:= (s2-d)/2;
count:= count+2;
tres:= tres, [c, b];
nt:= nt+1;
od;
if nt =1 then tres:= [tres]
else tres:= sort([tres], (a, b) -> evalb(a[2] < b[2]));
fi;
Res:= Res, op(map(op, tres));
od:
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MATHEMATICA
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r = Flatten[ Union@ Table[ If[ IntegerQ[ Sqrt[b + c]] && IntegerQ[ Sqrt[b*c]], {b, c}, Sequence @@ {}], {b, 10000}, {c, b, 10000}], 1]; Take[ Sort[r, #1[[1]] #1[[2]] < #2[[1]] #2[[2]] &], 36] // Flatten (* Robert G. Wilson v, Aug 03 2015 *)
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PROG
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(PARI) lista(nn) = {for (n=1, nn, sq = n^2; fordiv(sq, d, if ((d <= n) && issquare(d + sq/d), print1(d, ", ", sq/d, ", ")); ); ); } \\ Michel Marcus, Aug 03 2015
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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