OFFSET
2,1
COMMENTS
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
LINKS
D. Acu, On a diophantine equation 2^x + 5^y = z^2, General Mathematics Vol. 15, N° 4 (2007), 145-148.
A. Suvarnamani, A. Singta, S. Chotchaisthit, On two diophantine equations 4^x + 7^y = z^2 and 4^x + 11^y = z^2, Science and Technology RMUTT Journal, Volume 1 (2011), Number 1 : pp. 25 - 28.
MAPLE
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:
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Jan 20 2014
STATUS
approved