Given a continuous, unbounded function ( f: \mathbb{R} \to \mathbb{R} ), we need to determine if there exists an ( x \in \mathbb{R} ) such that the set ( { |f(nx)| \, | \, n \in \mathbb{Z} } ) is unbounded. Thus, the answer is (\boxed{Yes}).
Month: January 2025
Vieta Cubic 01a in Pari/gp
A finite fields probable prime test using Galois conjugates and Vieta’s formulas
I’ve been trying to extend the cubic Frobenius test to a quartic test, and it’s been challenging. I got an idea for a primality test that isn’t fast, but is interesting in any case. Let $n$ be a probable prime, and $f$ be a monic 4th degree polynomial such that $f$ has no roots in… Continue reading A finite fields probable prime test using Galois conjugates and Vieta’s formulas
Cubic Frobenius test statistics from 1.0e13 to 2.0e13
For composites in the range 1.0e13 to 2.0e13, there are two instances of type 2 (passing Congruence 1 only) and no instances showing they passed either Congruence 2 or Congruence 3. For the primes in the range, they all pass congruences 1 to 5. Processing block 1000: 10999000000000 to 11000000000000Type 0: 233313317052Type 1: 0Type 2:… Continue reading Cubic Frobenius test statistics from 1.0e13 to 2.0e13
When might one expect a Fermat-Lucas pseudoprime?
The Baillie-PSW primality test consists of a strong Fermat base $2$ test and a standard or strong Lucas sequences test. For an odd number $n$, the Fermat base $2$ test consists in checking whether the congruence $2^{n-1} \equiv 1 \pmod n$. A composite $n$ that passes the Fermat base $2$ test is known as a… Continue reading When might one expect a Fermat-Lucas pseudoprime?
Random subset heuristics applied to pseudoprimes
Suppose there are N tickets in an urn of which m < N are winning tickets. You have n tries to get a winning ticket. The tickets are replacedin the urn after each draw. Assume m and n are much smaller than N. Show that the probability of failure p_fail is approximated by: p_fail ~=… Continue reading Random subset heuristics applied to pseudoprimes
A naive implementation of Erdos’ Carmichael number construction
I’ve been reading some about the Erdos heuristic to construct Carmichael numbers, first described by Erdos in 1956. It is based on Korselt’s criterion, that N is a Carmichael number if N is composite, squarefree and such that if p|N, then (p-1)|(N-1). The best reference I’ve found to understand the basics of Erdos’s heuristic is… Continue reading A naive implementation of Erdos’ Carmichael number construction