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A236228
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Number of solutions (x,y,z) to the Diophantine equation 2^x + p^y = z^2 where p = prime(n).
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0
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3, 2, 3, 1, 1, 6, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 3, 2, 1, 1, 1, 1, 3, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2
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OFFSET
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2,1
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COMMENTS
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These numbers were obtained experimentally using Maple with p = 3,5,...,prime(1000)=7919 and x, y = 0,1,2,...,30. We find x,y <= 23.
p > 2 because for p = 2 the number of solutions is infinite.
The trivial solution (x,y,z) = (3,0,3) exists for all prime p.
The following table gives the first values (p, x, y, z)
+----+-----+----+-----+
| p | x | y | z |
+----+-----+----+-----+
| 3 | 0 | 1 | 2 |
| 3 | 3 | 0 | 3 |
| 3 | 4 | 2 | 5 |
| 5 | 2 | 1 | 3 |
| 5 | 3 | 0 | 3 |
| 7 | 1 | 1 | 3 |
| 7 | 1 | 1 | 3 |
| 7 | 5 | 2 | 9 |
| 11 | 3 | 0 | 3 |
.......................
The maximum x should be at least 30 because of the identity 2^30 + (2^16+1)^1 = (2^15+1)^2 involving the conjectured greatest prime Fermat number 2^16+1 = 65537. - Jean-François Alcover, Jan 24 2014
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LINKS
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MAPLE
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with(numtheory):nn:= 30:for n from 2 to 90 do:c:=0:p:=ithprime(n):for x from 0 to nn do:for y from 0 to nn do:z:=sqrt(2^x+p^y): if z=floor(z)then c:=c+1:else fi:od:od: printf(`%d, `, c):od:
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MATHEMATICA
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xm = 30; f[n_] := With[{p = Prime[n]}, Table[Table[{x, y, Sqrt[2^x + p^y]}, {y, 0, Log[p, Max[1, 2^xm - 2^x]]}], {x, 0, Log[2, 2^xm]}] // Flatten[#, 1]& // Union]; sol[n_] := Select[f[n], IntegerQ[# // Last]&]; a[n_] := sol[n] // Length; Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Jan 24 2014 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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