Re: Computability

On Jan 10, 11:16 am, quasi <qu...@xxxxxxxx> wrote:

You claimed to have checked it up to z = 256, but apparently you
missed the following counterexample:

  (x,y,z) = (74,129,143) which has signature (6,3,3).

Wow. At first I thought this was a candidate triple, because for this
triple (x,y,z) the following are indeed all true:

x < y < z < x+y
(x,y) = (y,z) = (z,x) = 1
o(x/y,z)/2 = o(z/x,y) = o(z/y,x) = p in Po, the odd primes ( o = 3
here)

but then I went back and checked more rigorously and checked
probablilities, too.

at probability = about 29%, x < y < z < x+y
at probability = about 21%, (x,y) = (y,z) = (z,x) = 1 independently
at probability = << 1%, o = odd prime independently of the above
at probability so far zero, (x/y)^p == -1 but only in combination with
the above 3.

and that last is the test for which this fails. As one might expect,
with such a tiny probability that might slip through, so yes, while
it has the signature, it doesn't pass that one test:

(x/y)^p == 143 - 105 == -38, not -1 which would be 142 mod 143

This comes from the first derivation I made in modular arithmetic, way
back, gosh, it was mentioned near the beginning of this thread, and in
2004, I think:

From

x^p + y^p = z^p we have
x^p + y^p == 0 mod z, and so
x^p == -(y^p) mod z, and
(x^p)/(y^p) == -1 mod z, and
(x/y)^p == -1 mod z, where
/y means the multiplicative inverse of y in the reduced residues of z,
that is, in what my text calls Zz*.

Darn. For a moment, there, I thought I'd finally be able to drop this
search! Only around 8 times as many triples to check to limit = 256,
but sooo many to limit = 65535. Dang it, Chip Eastham, you rascal,
what were we/I thinking!?

Thanks to quasi for the smallest triple with the right prime
signature. Well done!

(74, 129, 143) Really, that is good work. To limit = 143, my program
checked 1,415 triples coprime and with the inequality, from 343,000
generated triples. Wow.

Doug

P.S. The smallest triple with the inequality and the coprimality is
the classic (3,4,5)...

P.P.S. I think another condition is x + y == z mod p. I am working to
understand that one....
.

Relevant Pages

  • Re: Computability
    ... missed the following counterexample: ... signature. ... checked 1,415 triples coprime and with the inequality, from 343,000 ...
    (sci.math)
  • Re: Computability
    ... missed the following counterexample: ... signature. ... checked 1,415 triples coprime and with the inequality, from 343,000 ...
    (sci.math)
  • Re: Computability
    ... missed the following counterexample: ... signature. ... checked 1,415 triples coprime and with the inequality, from 343,000 ...
    (sci.math)
  • Re: Computability
    ... but we are mising the point. ... inequality, without loss of generality, and the signature, but the ... signature would have four parts, ... inequality, the coprimality, and the signature, applying Bayesian ...
    (sci.math)