Revision history [back]
 | 1 | initial version |
This is not a bug.
sage: var('x,y,z')
sage: ((x+y+z)^5).expand()
x^5 + 5*x^4*y + 10*x^3*y^2 + 10*x^2*y^3 + 5*x*y^4 + y^5 + 5*x^4*z + 20*x^3*y*z + 30*x^2*y^2*z + 20*x*y^3*z + 5*y^4*z + 10*x^3*z^2 + 30*x^2*y*z^2 + 30*x*y^2*z^2 + 10*y^3*z^2 + 10*x^2*z^3 + 20*x*y*z^3 + 10*y^2*z^3 + 5*x*z^4 + 5*y*z^4 + z^5
Note the divisibility by 5 of all terms which are "missing" in your characteristic $p=5$ calculation.
This is a consequence of the Freshman's dream: $(x+y)^p = x^p + y^p$ (just apply it twice).
It looks too good to be true, but it really is true in characteristic $p$ (the proof is in the linked article).
| | 2 | No.2 Revision |
This is not a bug.
sage: var('x,y,z')
sage: ((x+y+z)^5).expand()
x^5 + 5*x^4*y + 10*x^3*y^2 + 10*x^2*y^3 + 5*x*y^4 + y^5 + 5*x^4*z + 20*x^3*y*z + 30*x^2*y^2*z + 20*x*y^3*z + 5*y^4*z + 10*x^3*z^2 + 30*x^2*y*z^2 + 30*x*y^2*z^2 + 10*y^3*z^2 + 10*x^2*z^3 + 20*x*y*z^3 + 10*y^2*z^3 + 5*x*z^4 + 5*y*z^4 + z^5
Note the divisibility by 5 of all terms which are "missing" in your characteristic $p=5$ calculation.
This is a consequence of the Freshman's dream: $(x+y)^p = \equiv x^p + y^p$ (just y^p \pmod p$; just apply it twice).twice.
It looks too good to be true, but it really is true in characteristic $p$ (the proof is in the linked article).
