I’ll be looking for numbers **p** such that **p = (p + 1)^k + (p + 1)^k – (p + 2)^k**

If **k = 2**, then

**p = (p + 1)^2 + (p + 1)^2 – (p + 2)^2 **

-p^2 + p + 2 = 0

Solutions: p = -1 p = 2

2 = 3 + 3 – 4 and 2 = 3^2 + 3^2 – 4^2

If **k = 3**, then

**p = (p + 1)^3 + (p + 1)^3 – (p + 2)^3 **

-p^3 + 7p + 6 = 0

Solutions: p = -2 p = -1 p = 3

The only acceptable solution: p = 3

3 = 4 + 4 – 5 and 3 = 4^3 + 4^3 – 5^3

For any k > 3, there does not exist an integer p such that

p = (p + 1)^k + (p + 1)^k – (p + 2)^k

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## About benvitalis

math grad - Interest: Number theory

Thanks.