Research in Scientific Computing in Undergraduate Education

Catalan pseudoprimes

As a special case of Fermat’s little theorem, if p is an odd prime number then 2^p-2 is divisible by p, or, in the language of congruences, 2^p \equiv 2 (mod \phantom{.}p).

The converse of this is not true: if 2^p \equiv 2 (mod \phantom{.}n) where n is a positive integer, it does not follow that n is prime. For example n could be 561 = 3\times11\times 17. There are infinitely many such n.

Composite integers that pass a test for prime numbers are commonly known as pseudoprimes. Of course whether a composite number is regarded as a pseuodprime depends on the particular prime number property we are using.

Another property of primes involves the Catalan numbers. These are the numbers C_n defined by the recurrence relation C_0=1 and C_{n+1}=C_0C_n+C_1C_{n-1}+\ldots+C_{n-1}C_1+C_nC_0 \textrm{ for } n \geq 0.

The Catalan numbers can also be defined recursively by C_{n+1}=\frac{2(2n+1)}{n+2}C_n \textrm{ for } n\geq 0.

The Catalan numbers are the coefficients of x in the series expansion of \frac{1-\sqrt{1-4 x}}{2 x}=1 + x + 2x^2 + 5x^3 + 14x^4 + 42x^5 + 132x^6 + 429x^7 + 1430x^8 + 4862x^9 + 16796x^10+\ldots

The Catalan numbers up to C_{25} are

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452.

Clearly C_n grows rapidly with n. In fact C_n \sim \frac{4^n}{n^{3/2}\sqrt{\pi}} in the sense that the ratio of C_n to \frac{4^n}{n^{3/2}\sqrt{\pi}} approaches 1 as n \to \infty:

Theorem 2 of Aebi & Cairns (see below) provides a property of prime numbers in terms of Catalan numbers:

If p is an odd prime, then (-1)^{\frac{p-1}{2}} .C_{\frac{p-1}{ 2} }\equiv 2 (mod \phantom{.}p)

On the other hand, there are numbers other than primes that also have this property, for example p=5907 = 3\times11\times179.

Aebi & Cairns call a composite number n a Catalan pseudoprime if (-1)^{\frac{n-1}{2}} .C_{\frac{n-1}{ 2} }\equiv 2 (mod \phantom{.}n). Thus, 5907 is a Catalan pseudoprime.

Currently the only known Catalan pseudoprimes are:

  • 5907=3 \times 11\times179
  • 1093^2 = 1093 \times 1093
  • 3511^2 = 3511 \times 351

The object of this project is to use computational methods to find other Catalan pseudoprimes in order to help determine how frequently they occur.

The following two lines of Mathematica® code:

n = 1;
While[Or[Mod[(-1)^((n - 1)/2)*CatalanNumber[(n - 1)/2], n] != 2, PrimeQ[n] == True], n = n + 2]

found 5907 in 1.86 seconds.

Beyond that things are much more difficult.

A related question

The Catalan numbers are related to the middle binomial coefficients \left(\begin{array}{c c} n-1 \\ \ (n-1)/2 \end{array}\right) . When n is odd, say n =2k+1, we have \left(\begin{array}{c c} n-1 \\ \ (n-1)/2 \end{array}\right) = \left(\begin{array}{c c} 2k \\ \ k \end{array}\right) . A theorem of Erdös (see Finch, 2007, below) shows that almost always the middle binomial coefficient \left(\begin{array}{c c} 2k \\ \ k \end{array}\right) is divisible by high powers of small primes. Let g(k) denote the smallest odd prime dividing \left(\begin{array}{c c} 2k \\ \ k \end{array}\right). How does g(k) vary with k? A plot of g(k) \textrm{ versus } k \textrm{ for } 2\leq k\leq 10000 is shown below.

This plot indicates that although g(k) =3 commonly, there are values of k for which g(k) becomes much larger. In light of the theorem of Erdös cited above, a fundamental question is whether the numbers g(k) stay bounded as k increases.

The plot below shows the running average of the values of g(k). It indicates that, possibly, lim_{k\to\infty}\frac{1}{k}(g(2)+\ldots+g(k))=3.

Behavior of a number-theoretic function

The function c(k) :=(-1)^k \left(\begin{array}{c c} 2k \\ \ k \end{array}\right)\phantom{.}(mod \phantom{.} 2k+1) takes the value 1 exactly when 2k+1 is prime or a Catalan pseudoprime. How does c(k) behave as a function of k?

A plot of c(k) \textrm{ for } 1 \leq k \leq 5000, below, indicates that c(k) is bounded above by a linear function of k but has wide variation:

A plot for 1\leq k \leq 20000 shows similar behavior, but appears more dense on the same scale:

This is more or less what we would expect if the values of c(k) distributed themselves relatively uniformly across the remainders mod \phantom{.} 2k+1. A plot of the running average \mu(k):=\frac{1}{k}\sum_{i=1}^k c(i), and of the running standard deviation \sigma(k):=\sqrt{\frac{1}{k}\sum_{i=1}^k(c(i)-\mu(k))^2}, indicates a linear increase with k:

Running average of c(k)

Running average of c(k)

Running standard deviation of c(k)

Running standard deviation of c(k)

How does the function ac(k) :=(\left(\begin{array}{c c} 2k \\ \ k \end{array}\right) - (-1)^k )/(2k+1) \phantom{.}(mod \phantom{.} 1) behave as a function of 2k+1? The plot below indicates that the values are not uniformly distributed:

A blow-up around the value \frac{1}{3} shows the following behavior:

Similar behavior holds around the value \frac{2}{3}.

Similar, but weaker, behavior occurs around the values \frac{1}{5} \textrm{ and } \frac{1}{7}:

References & readings

One Response

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  1. Bernard Casavant said, on April 9, 2013 at 11:59 pm

    The last catalan factorization should be 3511 x 3511 instead 3511 x 351


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