## Re: "wriggly" pseudoprime puzzle

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• Dear David, ... Please give a hint how you produce these impression rate, some mathematical background would be very nice, perhaps also for others. I would
Message 1 of 50 , Sep 5 1:24 PM
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Dear David,

> > 1. p is an odd positive integer
> > 2. a is an integer with kronecker(a,p) = -1
> > 3. a^((p-1)/2) = -1 mod p
> > 5. Mod(1+x,x^2-a)^p = 1-x mod p
>
> > Puzzle: Find a pseudoprime, p,
> > with a false witness, a,
> > that fools (1), (2), (3), (5) and has
> > znorder(Mod(a,p)) > 3414.
>
> Comment: The gremlins can generate false witnesses
> for this puzzle at a rate in excess of 50 kHz,
> finding more than 600,000,000 in less than 3 hours.

Please give a hint how you produce these impression
rate, some mathematical background would be very nice,
perhaps also for others.

I would like to participate in the calculation,
but how do you generate pseudoprimes for which (1), (2), (3) and (5)
are o.k. ?

By the way, is there a mathematical book, about
which somebody could recommand ?

> Hint: All of these false witnesses fail my gremlin-trap:
>
> > 4. gcd(a^6-1,p) = 1

Pure chance ;-)

Nice Greetings from the primes
Bernhard
• ... [4] is meaningless, as it stands. You should write a double mod: 4. (1+x)^p = 1-x mod(x^2-a,p) ... There is no reason whatsoever to believe that [1] to [4]
Message 50 of 50 , Sep 29 4:14 AM
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"bhelmes_1" <bhelmes@...> wrote:

> 1. let a jacobi (a, p)=-1
> 2. let a^(p-1)/2 = -1 mod p
> 3. a^6 =/= 1 mod p
> 4. (1+sqrt (a))^p = 1-sqrt (a)

[4] is meaningless, as it stands.
You should write a double mod:

4. (1+x)^p = 1-x mod(x^2-a,p)

> 1. Is it possible that there are other exceptions

There is no reason whatsoever to believe that
[1] to [4] establish that p is prime. Morevoer,
some folk believe that, for every epsilon > 0,
the number of pseudoprimes less than x may
exceed x^(1-epsilon), for /sufficiently/ large x.

> 2....
> there is a cyclic order ...

> 3....
> there is a cyclic order ...

The group of units (Z/nZ)* is /not/ cyclic
if n has at least two distinct odd prime fators.

David
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