From strass@godot.dfi.uni-duesseldorf.de  Mon Apr  3 14:49:47 2000
To: Michel Nguyen-The <mnguyen@lix.polytechnique.fr>
Subject: Re: Binomial Fermat
References: <200004010924.LAA09027@pam.polytechnique.fr>
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Status: RO

In a recent paper my colleague Helmut Finner and myself
considered the uniformly most powerful unbiased (UMPU)
one-sided test for the comparison of two proportions based
on sample sizes  n1  and  n2 , i.e., the randomized version
of Fisher's exact one-sided test.
We showed  that the power function of this test can coincide
on the entire parameter space with the power function of the
UMPU test based on sample sizes n1 + 1 and  n2  for certain
levels and values of n1 and n2. The characterization of all
such cases with identical power functions is closely related
to the triangular version of the Fermat equation.

(1) binomial(x,n) + binomial(y,n) = binomial(z,n)

The existence of a solution  may be described via two urn experiments.
Suppose we have two urns.  Urn 1 with  x  white and  z - x black
balls and urn 2  with  y  white and  z - y  black balls.
Both  experiments consists in drawing exactly  n   balls from
urn 1 and n balls from urn 2 (n<=min(x,y,z)).
Let E1 denote the event of drawing no black ball from urn 1 and
let E2 denote the event of drawing at least one black ball from
urn 2. The question is how to choose the numbers x, y, z and n,
such that the events E1 and E2 have the same probability.,
i.e., binomial(x,n) / binomial(z,n)  = 1 - binomial(y,n) / binomial(z,n)

which is obviously equivalent to (1).

Harborth (1988) called (1) a Fermat-like binomial equation.
Fraenkel (1971) considered a more general class of diophantine
equations. However, (1) has at least one solution for each  n,
namely x = y = 2n-1, z = 2n . This has already been mentioned
in Fraenkel (1971). The question is, whether there
exist further solutions for n >=2. For n = 2 Khatri, M. N. (1955),
n= 3 Sierpi\`{n}ski (1962)
proved that the number of solutions of (1) is infinite, confer also the
remarks in Fraenkel (1971) and Wunderlich (1962)  on this topic.
For n >= 4, the number of further solutions seems to be rather small
although Harborth (1988) proved that there exist infinitely many
n's with further solutions. Fraenkel (1971) conjectures that the number
of solutions is finite for any fixed n > 3. Numerical calculations have
shown, that there exist only seven solutions (n,x,y,z) for 4<= n <= 500,
x<=y, 1<=z-n<= 500 namely
(4, 132, 190, 200), (6, 14, 15, 16), (6, 13, 13, 15),
(35, 118, 118, 120), (40, 103, 104, 105), (204, 695, 695, 697)
and (273, 713, 714, 715).
We found no further solution for  n = 4  for  z <= 126937,
for  n = 5 , 6   for  z <=10000. As far as we know it is not known
whether the set of solutions of (1) for fixed  n >= 4  is finite.

EIS-sequences of solutuions (which are partially slightly shifted)
for  n = 2,3,4  of  (1) are  A012132, A002311, A020329.

References

Fraenkel, A. S. (1971).
Diophantine equations involving generalized triangular and tetrahedral
numbers.
Computers in Number Theory,
Proc. Atlas Sympos. No. 2, Oxford 1969, 99-114.

Harborth, H. (1988).
Fermat-like binomial equations.
In: A.N. Philippou et al. (eds.), Applications of Fibonacci Numbers,
Kluwer Academic Publ., 1 1-5.

Khatri, M. N. (1955).
Triangular numbers and Phythagorean triangles.
Scripta Math  21, 94.

Sierpi\`{n}ski, W. (1962).
Sur une propri\`{e}t\`{e} des nombres t\`{e}tra\`{e}draux.
Elem. Math. 17, 29-30.

Wunderlich, M. (1962).
Certain properties of pyramidal and figurate numbers.
Math. Comp. 16, 482-486.



If you like we can send our original paper as dvi file.
The paper is submitted for publication but not exepted yet.
If you like you can put a link to the paper on your home page.

Regards
Klaus Strassburger & Helmut Finner