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A236228 Number of solutions (x,y,z) to the Diophantine equation 2^x + p^y = z^2 where p = prime(n). 0
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=2..88.

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

Sequence in context: A106693 A107335 A200223 * A082391 A283977 A248579

Adjacent sequences:  A236225 A236226 A236227 * A236229 A236230 A236231

KEYWORD

nonn

AUTHOR

Michel Lagneau, Jan 20 2014

STATUS

approved

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Last modified February 28 01:38 EST 2020. Contains 332319 sequences. (Running on oeis4.)