Catalan's Conjecture

The conjecture made by Belgian mathematician Eug�nie Charles Catalan in 1844 that 8 and 9 (2³ and 3²) are the only consecutive powers (excluding 0 and 1). In other words,

3² - 2³ = 1

x^a - y^b = ±1

The conjecture was finally proved by Preda Mihailescu in a manuscript privately circulated on April 18, 2002.  The proof has now appeared in print (Mihailescu 2004) and is widely accepted as being correct and valid (Daems 2003, Mets�nkyl� 2003), according to Mathworld. 

A Special Case of Catalan's Conjecture

This special case can be phrased this way: "no power of 2 differs by 1 from any power other than 3²".

Suppose 2^a = (2k+1)^b + 1.

If b is odd then 2^a = (2k+1)^b+1 = (2k+2)((2k+1)^b-1 - (2k+1)^b-2 + ... + 1)
The second factor, (2k+1)^b-1 - (2k+1)^b-2 + ... + 1, is an odd number greater than 1, which can't be a factor of 2^a, so b can't be odd.

If b is even then (2k+1)^b = 1 mod 4, so
2^a = (2k+1)^b+1 = 2 mod 4.
But 2^a = 0 mod 4 (because a>1), a contradiction.

So there is no value of b such that 2^a = (2k+1)^b + 1.

Now suppose 2^a = (2k+1)^b - 1.

If b is odd then 2^a = (2k+1)^b-1 = (2k)((2k+1)^b-1 + (2k+1)^b-2 + ... + 1)
The second factor, (2k+1)^b-1 + (2k+1)^b-2 + ... + 1, is an odd number greater than 1, so b can't be odd.

If b is even then 2^a = (2k+1)^b-1 = (4k²+4k+1)^b/2-1 =
      (4k²+4k)((4k²+4k+1)^b/2-1 + (4k²+4k+1)^b/2-2 + ... + 1)
The first factor, (4k²+4k), is a power of 2, so all its factors must be powers of 2.  In particular, k and k+1 must both be powers of 2, which means k=1.

Since k=1, and b is even, we have 2^a = 3^b-1 = 9^b/2-1 = 8(9^b/2-1 + 9^b/2-2 + ... + 1)

If b≠2 and b/2 is odd, then 9^b/2-1+9^b/2-2+...+1 is odd and larger than 1, and thus can't be a factor of 2^a, so b/2 is even.

If b/2 is even, then 2^a = 3^b-1 = 81^b/4-1 = 80(81^b/4-1 + 81^b/4-2 + ... + 1), a multiple of 5, which is a contradiction.

That leaves only b=2, giving our familiar solution.

Another Special Case of Catalan's Conjecture

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Internet references

Catalan's Conjecture in Wolfram's Mathworld

Eug�nie Charles Catalan in Wolfram's Scienceworld

George Baloglou's "thirteen" page

Related pages in this website

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