# Number Theory

## 0912 Submissions

[3] **viXra:0912.0043 [pdf]**
*replaced on 21 Dec 2009*

### Imanol's Numbers

**Authors:** Imanol Pérez

**Comments:** 2 Pages.

Imanol's numbers are those that the sum of their digits is 2, 3, 5, 6, 8 or 9.

**Category:** Number Theory

[2] **viXra:0912.0040 [pdf]**
*submitted on 18 Dec 2009*

### Expansión (1/x+2/x.......+a/x)^{n}

**Authors:** Imanol Pérez

**Comments:** 2 Pages. In Spanish

Expansion of (1/x+2/x.......+a/x)^{n}

**Category:** Number Theory

[1] **viXra:0912.0030 [pdf]**
*submitted on 12 Dec 2009*

### Diophantine Equation 1^{N} + 2^{N} + ...+ (M 1)^{N} +M^{N} = (M + 1)^{N}

**Authors:** Arkoprobho Chakraborty

**Comments:** 13 pages.

Erdos had conjectured that the equation of the title had no solutions
in natural numbers except the trivial 1^{1} + 2^{1} = 3^{1}. Moser (1953) had
shown that there are no solutions for M+1 < 10^{106}. Butske et al (1993)
had further shown that there are no solutions for M+1 < 9.3x10^{6}. In
this paper I show that a solution to this equation cannot exist for any
value of M > 2 hence proving Erdos' conjecture. This is achieved using
elementary number theoretic methods employing congruences and well-known identities.

**Category:** Number Theory