According to Wikipedia, Beal's conjecture is a conjecture in number theory proposed by Andrew Beal in 1993. While investigating generalizations of Fermat's last theorem in 1993, Beal formulated the following conjecture:
If
Ax + By = Cz
where A, B, C, x, y and z are positive integers with x, y, z > 2, then A, B, and C have a common prime factor.
For a proof or counterexample published in a refereed journal, Beal initially offered a prize of $ 5,000 in 1997, rising to $50,000 over ten years, and now has been raised to $1,000,000 (as of June 6, 2013).
In finding a counterexample, with the help of the [dbo].[spt_values] table in the master database, a simple query can be used to identify different possible values for the six variables that will satisfy the equation.
DECLARE @MaxBase INT
DECLARE @MaxPower INT
SET @MaxBase = 10
SET @MaxPower = 10
SELECT A.[Number] AS A, B.[Number] AS B, C.[Number] AS C, x.[Number] AS x, y.[Number] AS y, z.[Number] AS z
FROM (SELECT CAST([Number] AS FLOAT) AS [Number]
FROM [master].[dbo].[spt_values]
WHERE [type] = 'P' AND [number] BETWEEN 1 AND @MaxBase) A
CROSS JOIN
(SELECT CAST([Number] AS FLOAT) AS [Number]
FROM [master].[dbo].[spt_values]
WHERE [type] = 'P' AND [number] BETWEEN 1 AND @MaxBase) B
CROSS JOIN
(SELECT CAST([Number] AS FLOAT) AS [Number]
FROM [master].[dbo].[spt_values]
WHERE [type] = 'P' AND [number] BETWEEN 1 AND @MaxBase) C
CROSS JOIN
(SELECT CAST([Number] AS FLOAT) AS [Number]
FROM [master].[dbo].[spt_values]
WHERE [type] = 'P' AND [number] BETWEEN 3 AND @MaxPower) x
CROSS JOIN
(SELECT CAST([Number] AS FLOAT) AS [Number]
FROM [master].[dbo].[spt_values]
WHERE [type] = 'P' AND [number] BETWEEN 3 AND @MaxPower) y
CROSS JOIN
(SELECT CAST([Number] AS FLOAT) AS [Number]
FROM [master].[dbo].[spt_values]
WHERE [type] = 'P' AND [number] BETWEEN 3 AND @MaxPower) z
WHERE POWER(A.[Number], x.[Number]) + POWER(B.[Number], y.[Number]) = POWER(C.[Number], z.[Number])
ORDER BY A.[Number], B.[Number], C.[Number]
Here's the first ten values for the six variables that satisfy the equation:
A B C x y z
--- --- --- --- --- ---
2 2 2 4 4 5
2 2 2 8 8 9
2 2 2 3 3 4
2 2 2 7 7 8
2 2 2 6 6 7
2 2 2 5 5 6
2 2 2 9 9 10
2 2 4 5 5 3
2 2 4 7 7 4
2 2 4 9 9 5
Unfortunately, no counterexample was identified using this query. Also, as the value of the x, y and z variables grow bigger, the result becomes incorrect. As an example, increasing the value of the @MaxPower to 20 yields the following incorrect results:
A B C x y z
--- --- --- --- --- ---
1 7 7 3 20 20
1 7 7 4 20 20
1 7 7 5 20 20
1 7 7 6 20 20
1 7 7 7 20 20
1 7 7 8 20 20
1 7 7 9 20 20
1 7 7 10 20 20
1 7 7 11 20 20
1 7 7 12 20 20