Mathematics Stack Exchange
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Questions tagged [pseudoprimes]

Pseudoprimes are composite numbers which pass some primality test - a property that is always true for prime numbers. This may be Fermat's Little Theorem for one base or many, or some other test.

Pseudoprimes are composite numbers that have some properties that every prime number has (that is, properties that might be used to test for primality).

For example, if $p$ is a prime number and $\gcd(b, p) = 1$, then, by Fermat's little theorem, $b^{p - 1} \equiv 1 \pmod p$. However, there are also composite numbers that satisfy this congruence for some coprime $b$ (these are called Fermat pseudoprimes) and composite numbers that satisfy this property for every coprime $b$ (these are called Carmichael numbers).

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Can a Mersenne number ever be a Carmichael number?

Can a Mersenne number ever be a Carmichael number? More specifically, can a composite number $m$ of the form $2^n-1$ ever pass the test: $a^{m-1} \equiv 1 \mod m$ for all intergers $a >1$ (Fermat's Test)? Cases potentially proved so far: (That are…
Dane Bouchie
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Are there any $3$-Fermat-pseudoprimes of the form $k^4+1$?

Suppose $k$ is a positive integer and $N:=k^4+1$ is composite. Can $N$ be a $3$-Fermat-pseudoprime; that is, can the congruence $$3^{N-1}\equiv 1 \mod{N}$$ hold ? I checked up to $k=10^8$ and found no example. Is this simple test in fact…
Peter
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If $b$ is even and not a power of two, can $b^4+1$ be a Fermat pseudoprime base 2?

The complete question is already in the title but we shall provide some motivation as well. We study generalized Fermat numbers defined by: $$\mathrm{GF}(n,b) = b^{2^n}+1$$ where $b$ and $n$ are natural numbers and we are interested in cases where…
Jeppe Stig Nielsen
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For which $n$ do $A+A^n+A^{n^2} \equiv 0 \pmod n$, $A^{n^2+n+1} \equiv 7\mathrm{I} \pmod n$ hold, A the companion matrix of $x^3-7x-7$?

Let $f=x^3-7x-7$. Suppose $p$ is a prime and $f$ is irreducible over $\mathbb{F}_p$, and let $K = \mathbb{F}_p[x]/(f)$, the field with $p^3$ elements. If we let $\alpha = x \pmod f$ then $\alpha$ is a root of $f$ in $K$. The map $\sigma: K \mapsto…
David Bernier
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Some heuristics about the Pisano Period, primes and Fibonacci primes. What reasons are behind them?

I started to read about the Pisano Period, $\pi(n)$, applied to the classic Fibonacci sequence and made some simple tests looking for possible properties of the sequence. I have observed the following ones, tested for the first 10000…
iadvd
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Was the number $3,317,044,064,679,887,385,961,981$ doublechecked?

In Wikipedia , Miller-Rabin-Test, it is mentioned that the smallest strong-Fermat pseudoprime to the prime-bases upto $41$ is $$3,317,044,064,679,887,385,961,981$$ Hence, every number smaller than this number is prime, if it passes the…
Peter
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Carmichael numbers of form $m^3+1$ and Ramanujan's $1729$

While researching for a post on tetranacci pseudoprimes I came across a list of Carmichael numbers, $$C_n = 561,\, 1105,\, 1729,\, 2465,\, 2821,\dots$$ Of course, Ramanujan's taxicab number $1729 = 12^3+1$ stood out. So I looked at other near-cubes…
Tito Piezas III
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What is the ratio of Carmichael pseduo-primes to true primes for $1$ to $n$? Or is it known?

Let $\pi(n)$ be the prime counting function. And let $\varphi(n)$ be the count of Carmichael pseudo-primes for $1$ to $n$. Is the ratio, $$\frac{\varphi(n)}{\pi(n)}$$ is known, as $n \to \infty$? I am asking this because of i want to know how much…
İbrahim İpek
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Do $n=2m+1$ and $\big(2^m\bmod(m\cdot n)\big)\in\{n+1,3n-1\}$ imply $n$ prime?

Do $n=2m+1$ and $\big(2^m\bmod(m\cdot n)\big)\in\{n+1,3n-1\}$ imply $n$ prime? Equivalently, for $n=2m+1$, do $2^m\equiv\pm1\pmod n$ and $2^m\equiv2\pmod m$ imply $n$ prime? Note: equivalence follows from the Chinese Remainder Theorem for $m>2$, and…
fgrieu
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Are there any primes that are never a factor of a Carmichael number?

Is there a prime number $p$ that $p > 2$, and in which $p$ is a never a factor of any Carmichael number $C_n$: (p ∤ $C_n$) After a quick glance at some Carmichael number factors, $p$ must be greater or equal to $53$.
Dane Bouchie
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What could explain this pseudoprime behavior of $26599 = 67 \times 397$?

If $f$ of degree $d$ is irreducible over $\mathbb{F}_p$, then $\mathbb{F}_p[x]/(f)$ is isomorphic to $\mathbb{F}_{p^d}$. Denoting $\mathbb{F}_p$ by $K$ and $\mathbb{F}_{p^d}$ by $L$, we have for $\alpha$ in $L$ the trace of $\alpha$: …
David Bernier
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Who can prove or disprove this conjecture?

Let $m$ be a positive integer such that $p:=8m^2+1$ is prime. Conjecture : We always have $$2^{2m^2}\equiv 1\pmod p$$ I could only establish $$2^{4m^2}\equiv 1\pmod p$$ following from Euler's criterion , but I could not rule out $2^{2m^2}\equiv…
Peter
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What is the fastest way to get the next Carmichael-number?

A Carmichael number is a composite number $N$, such that $a^{N-1}\equiv 1\mod N$ holds for every $a$ coprime to $N$. $N$ is a Carmichael number if $N$ is odd and squarefree $N$ has at least three distinct prime factors For each prime factor $p|N$…
Peter
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What type of pseudoprime does the largest known pseudoprime tend to be?

It is a well-known fact that the largest known prime number for several decades now has been a Mersenne prime, even though more and more of them have been found over the years and there have also been efforts to find other kinds of primes, like the…
Mr. Brooks
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Conjecture about Rabin-Miller pseudo prime test

I tested the Rabin-Miller pseudo prime algorithm using a single test value and found that the number of false calls depends on the size of the number to test, reducing to a (conjectured) negligible probability of error for very large numbers. A…
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