p(p(x))=p(x+1)

Solution for p(p(x))=p(x+1) equation:

Simplifying
p(p(x)) = p(x + 1)

Multiply p * x
p(px) = p(x + 1)

Multiply p * px
p2x = p(x + 1)

Reorder the terms:
p2x = p(1 + x)
p2x = (1 * p + x * p)
p2x = (1p + px)

Solving
p2x = 1p + px

Solving for variable 'p'.

Reorder the terms:
-1p + -1px + p2x = 1p + px + -1p + -1px

Reorder the terms:
-1p + -1px + p2x = 1p + -1p + px + -1px

Combine like terms: 1p + -1p = 0
-1p + -1px + p2x = 0 + px + -1px
-1p + -1px + p2x = px + -1px

Combine like terms: px + -1px = 0
-1p + -1px + p2x = 0

Factor out the Greatest Common Factor (GCF), 'p'.
p(-1 + -1x + px) = 0

Subproblem 1
Set the factor 'p' equal to zero and attempt to solve:

Simplifying
p = 0

Solving
p = 0

Move all terms containing p to the left, all other terms to the right.

Simplifying
p = 0
Subproblem 2
Set the factor '(-1 + -1x + px)' equal to zero and attempt to solve:

Simplifying
-1 + -1x + px = 0

Reorder the terms:
-1 + px + -1x = 0

Solving
-1 + px + -1x = 0

Move all terms containing p to the left, all other terms to the right.

Add '1' to each side of the equation.
-1 + px + 1 + -1x = 0 + 1

Reorder the terms:
-1 + 1 + px + -1x = 0 + 1

Combine like terms: -1 + 1 = 0
0 + px + -1x = 0 + 1
px + -1x = 0 + 1

Combine like terms: 0 + 1 = 1
px + -1x = 1

Add 'x' to each side of the equation.
px + -1x + x = 1 + x

Combine like terms: -1x + x = 0
px + 0 = 1 + x
px = 1 + x

Divide each side by 'x'.
p = x-1 + 1

Simplifying
p = x-1 + 1

Reorder the terms:
p = 1 + x-1
Solution
p = {0, 1 + x-1}