mystery sequence (easy?)

[N. J. A. Sloane]

so it is hard to tell which of these may be new.
Lattices: A006966 (unlabeled); A055512 (labeled)
   Mitch Harris and Don Knuth: A102894-A102897.
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To: seqfan at ext.jussieu.fr
Subject: Fwd: solution of equations
Date: Sat, 17 Mar 2007 21:28:50 -0400
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I just received the following, which I am forwarding to the list 
without further comment.

Franklin T. Adams-Watters

-----Original Message-----
From: bhmd95 at yahoo.fr
To: FrankTAW at Netscape.net
Sent: Sat, 17 Mar 2007 5:51 PM
Subject: solution of equations

   March 17, 2007

Dear Colleague;
   I have the honor to address to you to inform you that I am an 
Algerian researcher in Mathematics, born in 1969 at Ain-sefra (- 
w-Naâma) and that after a research which lasted more than ten (10) 
years I succeeded in solving some problems which one can pose in 
natural numbers.

   I wanted to share with you the results of my research.
  1/Solution of the equation:X^2+(X+K)^2=Y^2 with K is a prime number 
and K=2*m^2-1, m>=2
 ex: K=7,17,31,71,...
  the X values are given by the sequence defined by: 
a(n)=6*a(n-3)-a(n-6)+2K with:
 a(0)=0, a(1)=2m+1, a(2)=6*m^2-10m+4
 a(3)=3K, a(4)=6*m^2+10m+4, a(5)=40*m^2-58m+21.
 click here: http://www.research.att.com/~njas/sequences/A118674

  2/Solution of the equation:X^2+(X+K)^2=Y^2 with K is a prime number 
and K=p^2-2, p>=5
 ex: K=23,47,79,...
  the X values are given by the sequence defined by: 
a(n)=6*a(n-3)-a(n-6)+2K with:
 a(0)=0, a(1)=2p+2,  a(2)=3*p^2-10p+8
 a(3)=3K, a(4)=3*p^2+10p+8, a(5)=20*p^2-58p+42.
 click here: http://www.research.att.com/~njas/sequences/A118675

 3/Solution of the equation:(X-Y)^3-XY=0
  the X values are given by the sequence defined by: a(n)=n*(n+1)^2, 
with n is a natural integer.
 click here: http://www.research.att.com/~njas/sequences/A045991
  the Y values are given by the sequence defined by: b(n)=(n+1)*n^2, 
with n is a natural integer.
 click here: http://www.research.att.com/~njas/sequences/A011379

 4/The equation:(X-Y)^m-XY=0, with m is an even number, m>=2
  I proved that this type of equations has no solution of course without 
taking the solution (0,0) in consideration.click here: 
http://www.research.att.com/~njas/sequences/A045991

 5/The equation:(X-Y)^m-XY=0, with m is an odd number, m>=5
  I could not until now find solutions,also I could not prove that this 
type of equations has no solution.
 Best regards;
 Mohamed Bouhamida.(bhmd95 at yahoo.fr)

--------
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