Tuesday, September 14, 2010
JSH: Proof forward of core error
A remarkable problem in number theory, a "core error" is shown in a forward direction in order to foil obfuscation attempts by posters dedicated to protecting the error.
Consider the following expression in the ring of algebraic integers:
9(g_1(x) + 1)(g_2(x) + 2) = 9*P(x)
where g_1(0) = g_2(0) = 0, and P(x) is a quadratic with integer coefficients; therefore, trivially, the last coefficient of P(x) is 2.
It is trivial to prove that it may not always exist in the ring of algebraic integers.
Proof:
Let P(x) be irreducible over Q, that is, not have rational roots for P(x) = 0. Then by the distributive property, I have trivially:
(9g_1(x) + 9)(g_2(x) + 2) = 9*P(x)
which must always exist then if the original expression exists. But now consider functions f_1(x), and f_2(x), such that
f_1(x) = 9g_1(x), and f_2(x) = g_2(x) - 7, so:
9(g_1(x) + 1)(g_2(x) + 2) = (f_1(x) + 9)(f_2(x) + 9) = 9*P(x)
where again g_1(0) = g_2(0) = 0, and notice one of the f's equals 0, at x=0, and P(x) is again a quadratic with integer coefficients.
By symmetry f_1(x) and f_2(x) will be roots of the same monic polynomial, which must itself be irreducible over Q, but then just one of the f's has 9 as a factor, and it is well-known that in the ring of algebraic integers that just one root of a monic polynomial with integer coefficients irreducible over Q may not have a non-unit rational as a factor if ALL of the roots do not have it as a factor.
Therefore 9 is a factor of NEITHER of the f's in the ring of algebraic integers, and there is a contradiction.
Proof complete.
Going forward allows me to get away from arguments by posters claiming that the distributive property does not apply by directly using it in a key step.
Notice then that the ring of algebraic integers can be picky!
It will not allow under certain circumstance:
9(g_1(x) + 1)(g_2(x) + 2) = 9*P(x)
where g_1(0) = g_2(0) = 0, and P(x) is a quadratic with integer coefficients.
But that shows gaps in the numbers covered as, why can there not be numbers that exist that will fulfill those conditions?
Those gaps are what are important, as the lack of full coverage by the ring of algebraic integers is a "core error" with critical implications for number theoretic results over 100+ years.
Having people arguing to hide an error, fighting for years to defend an error, is not surprising given the human condition, but it does show that error in and of itself is not something that people will reject!
Sometimes they embrace it, instead.
Sometimes people love error. Error becomes their true love.
Consider the following expression in the ring of algebraic integers:
9(g_1(x) + 1)(g_2(x) + 2) = 9*P(x)
where g_1(0) = g_2(0) = 0, and P(x) is a quadratic with integer coefficients; therefore, trivially, the last coefficient of P(x) is 2.
It is trivial to prove that it may not always exist in the ring of algebraic integers.
Proof:
Let P(x) be irreducible over Q, that is, not have rational roots for P(x) = 0. Then by the distributive property, I have trivially:
(9g_1(x) + 9)(g_2(x) + 2) = 9*P(x)
which must always exist then if the original expression exists. But now consider functions f_1(x), and f_2(x), such that
f_1(x) = 9g_1(x), and f_2(x) = g_2(x) - 7, so:
9(g_1(x) + 1)(g_2(x) + 2) = (f_1(x) + 9)(f_2(x) + 9) = 9*P(x)
where again g_1(0) = g_2(0) = 0, and notice one of the f's equals 0, at x=0, and P(x) is again a quadratic with integer coefficients.
By symmetry f_1(x) and f_2(x) will be roots of the same monic polynomial, which must itself be irreducible over Q, but then just one of the f's has 9 as a factor, and it is well-known that in the ring of algebraic integers that just one root of a monic polynomial with integer coefficients irreducible over Q may not have a non-unit rational as a factor if ALL of the roots do not have it as a factor.
Therefore 9 is a factor of NEITHER of the f's in the ring of algebraic integers, and there is a contradiction.
Proof complete.
Going forward allows me to get away from arguments by posters claiming that the distributive property does not apply by directly using it in a key step.
Notice then that the ring of algebraic integers can be picky!
It will not allow under certain circumstance:
9(g_1(x) + 1)(g_2(x) + 2) = 9*P(x)
where g_1(0) = g_2(0) = 0, and P(x) is a quadratic with integer coefficients.
But that shows gaps in the numbers covered as, why can there not be numbers that exist that will fulfill those conditions?
Those gaps are what are important, as the lack of full coverage by the ring of algebraic integers is a "core error" with critical implications for number theoretic results over 100+ years.
Having people arguing to hide an error, fighting for years to defend an error, is not surprising given the human condition, but it does show that error in and of itself is not something that people will reject!
Sometimes they embrace it, instead.
Sometimes people love error. Error becomes their true love.
# posted by James Harris @ 09:54 
