Number theory

Prime clusters and Riecoin

Posted on by John

Prime clusters are sets of primes that appear as close together as is generally possible.

There is one pair of consecutive prime numbers, 2 and 3, but there cannot be any more: in any larger pair of consecutive numbers, one of the pair will be even. But there are a lot of twin primes, perhaps infinitely many, and so a prime cluster of size two is a pair of primes whose difference is 2.

How close together can a set of three primes be? The set {2, 3, 5} has diameter 3, i.e. the difference between the largest and smallest element is 3. And the set {3, 5, 7} has diameter 4. But in general the diameter of a set of three primes must be at least 6. If the smallest element is bigger than 3, then all the elements are odd, but they cannot be consecutive odd numbers or else one of them would be divisible by 3. But there are many prime clusters of diameter 6. For example, {13, 17, 19} and {37, 41, 43}.

Formal definition and motivation

There is some fuzziness in the discussion above regarding what is generally possible. This section will make our definitions more rigorous.

In general a pair of primes cannot be consecutive numbers because one of the pair must be even. Stated more abstractly, every pair of integers larger than {2, 3} will contain a complete residue class mod 2, i.e. one of the numbers will be congruent to 0 mod 2 and one of the numbers will be congruent to 1 mod 2.

Now let’s look at sets of three primes. The example {2, 3, 5} is exceptional because it contains a complete residue class mod 2, i.e. it contains even and odd numbers. The example {3, 5, 7} is exceptional because it contains a complete residue class mod 3.

We say that a set of k primes is a cluster if it does not contain a full residue class modulo any prime. We could say it does not contain a full residue class modulo any prime q ≤ k because trivially no set of k elements can have set of more than k elements. For example, the cluster {13, 17, 19} does not contain a full set of residues mod 2 or 3, but it also does not have a full set of residues mod 5, 7, 11, ….

We say a cluster is maximally dense if it has the minimum diameter for a cluster of its number of primes.

Informally, we will call a maximally dense cluster simply a cluster. A maximally dense prime cluster is also sometimes called a prime constellation.

Riecoin cryptocurrency

I’ve written several posts about the Primecoin cryptocurrency whose proof of work task is finding prime chains. The Riecoin cryptocurrency requires miners to find prime clusters.

Primecoin is far from a major cryptocurrency, with a market cap of around $2.3M. Riecoin is about four times smaller than Primecoin. There is another number-theoretic cryptocurrency, Gapcoin, whose market cap is about 10x smaller than that of Riecoin. Safe to say these three projects are of more interest to mathematicians than investment firms. All three of these prime-based cryptocurrencies were launched between 2013 and 2014.

A proof of work task should satisfy three criteria:

  1. Solutions should be difficult to find but easy to confirm.
  2. The time to find a solution should be roughly predictable.
  3. The difficulty should be adjustable.

Finding prime clusters requires a brute force search, but it’s easy to test whether a cluster has been found, and so the first property is satisfied.

The Hardy-Littlewood conjectures give an estimate of the difficulty in finding prime clusters of a given length. The difficulty can be adjusted by adjusting the required length. So the Hardy-Littlewood conjectures assist in satisfying the second and third properties. It does not matter if the conjectures are false somewhere out in asymptopia; they empirically give good estimates for the size of numbers used in mining Riecoin.

More about clusters

The definition of a maximally dense cluster depends on a function s(k) that says what the diameter of a cluster of k primes would need to be. We’ve said s(2) = 2 and s(3) = 6. More values of s are listed on the page for OEIS sequence A008407.

Related posts

Efficiently testing multiple primes at once

Posted on by John

The previous post looked at a technique for inverting multiple integers mod m at the same time, using fewer compute cycles than inverting each integer individually. This post will do something analogous for prime chains, revisiting a post from a few days ago about testing prime chains.

A prime chain is a sequence of primes in which each is twice its predecessor, plus or minus 1. In a Cunningham chain of the first kind, it’s always plus, and in a Cunningham chain of the second kind, it’s always minus.

Primecoin is a cryptocurrency that uses finding prime chains as its proof-of-work (PoW) task. The miner has a choice of finding one of three kinds of prime chain: a Cunningham chain of the first or second kind, or a bi-twin chain. The length of the necessary chain varies over time to keep the difficulty relatively constant. Other PoW blockchains do something similar.

Some people say that Primecoin has miners search for primes for PoW. That’s not quite right. Miners have to find a chain of medium-sized primes rather than finding one big prime. This leads to more predictable compute times.

There is a way to test a candidate Cunningham chain of the second kind all at once. Henri Lifchitz gives his algorithm here. Given a sequence of numbers

n1, n2, n3, …, nk

where ni = 2ni−1 − 1 for each i and  n0 = 1 mod 4, all the numbers in the sequence are probably prime if

2^{n_{k-1} - 1} = 1 \bmod n_0 n_1 n_2 \cdots n_k

For example, consider the chain

31029721, 62059441, 124118881

Note that 31029721 mod 4 = 1 and 31029721 = 2*15514861 − 1. The following code demonstrates that the numbers in the chain are probable primes because it prints 1.

n0 = 15514861
n1 = 2*n0 - 1
n2 = 2*n1 - 1
n3 = 2*n2 - 1
prod = n0*n1*n2*n3
print( pow(2, n2 - 1, prod) )

Next I wanted to try the algorithm on much larger numbers where its efficiency would be more apparent, as in the previous post. But when I did, the test returned a result other than 1 on a known Cunningham chain of the second kind. For example, when I change the first two lines of code above to

n1 = 49325406476*primorial(9811, False) + 1
n0 = (n1 + 1) // 2

the code returns a large result. I verified that each of the numbers in the chain are prime using Sympy’s isprime function.

Usually a probable prime test can have false positives but never a false negative. I haven’t looked at Lifschitz method closely enough to tell whether it can have false negatives, but the code above suggests it can.

Tighter bounds in the prime number theorem

Posted on by John

The most elementary form of the prime number theorem says that π(x), the number of prime numbers less than x, is asymptotically equal to x / log(x). That’s true, but a more accurate result says π(x) is asymptotically equal to li(x) where

\text{li}(x) = \int_0^x \frac{dt}{\log t}

Five years ago I wrote about a result that was new at the time, giving a bound on |π(x) − li(x)| for x > exp(2000). This morning I saw a result in a blog post by Terence Tao that says

\left| \pi(x) - \text{li}(x) \right| \leq 9.2211\, x\sqrt{\log(x)} \exp\left( -0.8476 \sqrt{\log(x)} \right)

for all x ≥ 2. The result comes from this paper.

The new bound has the same form as the bound from five years ago but with smaller constants.

Efficiently computing multiple modular inverses at once

Posted on by John

Suppose you have a large prime number M and you need to find the inverse of several numbers mod M.  Montgomery’s trick is a way to combine the computation of the inverses to take less time than computing the inverses individually. Peter Montgomery (1947–2020) came up with this trick in 1985.

We will illustrate Montgomery’s trick by inverting three numbers—a, b, and c—though the trick extends to any number of numbers. It is commonly used in cryptography.

Modular inverses are much slower to calculate than modular products, so doing fewer of the former and more of the latter is a good tradeoff. Montgomery’s method only calculates one modular inverse, regardless of how many numbers need to be inverted.

The idea is to directly invert the product of all the numbers and use multiplication to find the inverses of the individual numbers. In our case, we compute

x = ab
y = cy = abc
x−1 = cy−1
b−1 = ax−1
a−1 = bx−1

To show that this actually saves time, we’ll run some Python code to invert three random numbers modulo a very large prime, much larger than occurs in practice. The reason is to make the computation time longer and easier to demonstrate. In practice, Montgomery’s trick saves a little time off of a lot of calculations. Here we’ll save a lot of time off a handful of calculations.

import sys
import time
from secrets import randbelow

# extend the default maximum integer size
sys.set_int_max_str_digits(100000)

# the 32nd Mersenne prime
M = 2**756839 - 1

def simple(a, b, c, M):
    return [pow(x, -1, M) for x in [a, b, c]]

def montgomery(a, b, c, M):
    x = a*b % M
    y = x*c % M
    yinv = pow(y, -1, M)
    cinv = x*yinv % M
    xinv = c*yinv % M
    binv = a*xinv % M
    ainv = b*xinv % M
    return [ainv, binv, cinv]
    
a = randbelow(M)
b = randbelow(M)
c = randbelow(M)

start = time.perf_counter()
result = simple(a, b, c, M)
elapsed = time.perf_counter() - start
print(elapsed)

start = time.perf_counter()
result = montgomery(a, b, c, M)
elapsed = time.perf_counter() - start
print(elapsed)

When we ran this, the direct approach took 121.8 seconds, and Montgomery’s trick took 47.6 seconds.

Related posts

Primecoin primality test

Posted on by John

When I wrote about how Primecoin uses prime chains for proof of work, I left out a detail.

To mine a new Primecoin block, you have to find a prime chain of specified length that starts with a number that is a multiple of the block header hash. According to the Primecoin whitepaper

Another important property of proof-of-work for cryptocurrency is non-reusability. … To achieve this, the prime chain is linked to the block header hash by requiring that its origin be divisible by the block header hash.

So given a hash h, you have to find k such that kh is the origin of a prime chain, where “origin” is defined in footnote [2] here.

Strictly speaking, the primes in a Primecoin prime chain are probable primes. Someone verifying a Primecoin blockchain will be satisfied that the block is authentic if the necessary numbers are prime according to the test used by the Primecoin software. Whether the numbers are actually prime is irrelevant.

Probabilistic primality tests are much more efficient than deterministic tests, and most applications requiring primes, such as RSA encryption, actually use probable primes. If a number is rejected as a probable prime, it’s certainly not a prime. If it is accepted as a prime, it very like is prime. More on probable primes here.

If you write your own Primecoin mining software using a different primality test, that’s fine as long as you actually find primes. But if one time you find pseudoprimes, composite numbers that pass your primality test, then your block will be rejected unless your pseudoprimes also pass the Primecoin software’s primality test.

Primecoin’s primality test

So what is Primecoin’s (probable) primality test? We quote the Primecoin whitepaper:

The classical Fermat test of base 2 is used together with Euler-Lagrange-Lifchitz test to verify probable primality for the prime chains. Note we do not require strict primality proof during verification, as it would unnecessarily burden the efficiency of verification. Composite number that passes Fermat test is commonly known as pseudoprime. Since it is known by the works of Erdös and Pomerance that pseudoprimes of base 2 are much more rare than primes, it suffices to only verify probable primality.

Fermat’s test

The Fermat test with base b says to test whether a candidate number n satisfies

bn − 1 = 1 mod n.

If n is prime, the equation above will hold. The converse is usually true: if the equation above hold, n is likely to be prime. There are exceptions, such as b = 2 and n = 561 = 3 × 11 × 17.

Euler-Lagrange-Lifchitz test

But what is the Euler-Lagrange-Lifchitz test? The whitepaper links here, a page by Henri Lifchitz that gives results generalizing those of Leonard Euler and Joseph-Louis Lagrange.

Testing 2p + 1

Suppose p is an odd prime. Then the Euler-Lagrange test says that if p = 3 mod 4, then q = 2p + 1 is also prime if and only if

2p = 1 mod q.

Lifchitz gives three variations on this result. First, if p = 1 mod 4, then q = 2p + 1 is also prime if and only if

2p = −1 mod q.

Together these two theorems give a way to test with certainty whether 2p + 1, i.e. the next term in a Cunningham chain of the first kind, is prime. However, the test is still probabilistic if we don’t know for certain that p is prime.

Testing 2p − 1

The last two results of Lifchitz are as follows. If p and q = 2p − 1 are prime, then

2p − 1 = 1 mod q

if p = 1 mod 4 and

2p − 1 = −1 mod q

if p = 3 mod 4.

The converses of these two statements give probable prime tests for q if we know p is prime.

So we can verify a (probable) prime chain by using Fermat’s test to verify that the first element is (probably) prime and use the Euler-Lagrange-Lifchitz test that the rest of the numbers in the chain are (probably) prime.

Related posts

Bi-twin prime chains

Posted on by John

I mentioned bi-twin prime chains in the previous post, but didn’t say much about them so as not to interrupt the flow of the article.

A pair of prime numbers are called twins if they differ by 2. For example, 17 and 19 are twin primes.

A bi-twin chain is a sequence of twin primes in which the average of each twin pair is double that of the preceding pair. For example,

5, 7, 11, 13

is a bi-twin chain because (5, 7) and (11, 13) are twin pairs of primes, and the average of the latter pair is twice the average of the former pair.

There is some variation in how to describe how long a bi-twin chain is. We will define the length of a bi-twin chain to be the number prime pairs it contains, and so the chain above has length 2. The BiTwin Record page describes a bi-twin chain of length k as a chain with k − 1 links.

It is widely believed that there are infinitely many twin primes, but this has not been proven. So we don’t know for certain whether there are infinitely many bi-twin prime chains of length 1, much less chains of longer length.

The bi-twin chain of length three with smallest beginning number is

211049, 211051, 422099, 422101, 844199, 844201

Bi-twin records are expressed in terms of primorial numbers. I first mentioned primorial numbers a few days ago and now they come up again. Just as n factorial is the product of the positive integers up to n, n primorial is the product of the primes up to n. The primorial of n is written n#. [1]

The longest known bi-twin chains start with

112511682470782472978099, 112511682470782472978101

and has length 9.

We can verify the chains mentioned above with the following Python code.

def bitwin_length(average):
    n = average
    c = 0
    while isprime(n-1) and isprime(n+1):
        c += 1
        n *= 2
    return c

for n in [6, 211050, 112511682470782472978100]:
    print(bitwin_length(n))

[1] There are two conventions for defining n#. The definition used here, and on the BiTwin Record page, is the product of primes up to n. Another convention is to define n# to be the product of the first n primes.

The priomorial function in Sympy takes two arguments and will compute either definition, depending on whether the second argument is True or False. The default second argument is True, in which case primorial(n) returns the product of the first n primes.

Prime chains

Posted on by John

The title of this post has a double meaning. We will look at chains in the sense of number theory and in the sense of cryptocurrency, i.e. Cunningham chains and blockchains, that involve prime numbers.

Cunningham chains

A chain of primes is a sequence of prime numbers in which each is almost double its predecessor. That is, the next number after p is 2p ± 1.

In a Cunningham chain of the first kind, the successor of p is 2p + 1. For example,

41, 83, 167.

In a Cunningham chain of the second kind, the successor of p is 2p − 1. For example,

19, 37, 73.

Two questions come up immediately. First, are there infinitely many Cunningham chains? Second, how long can a Cunningham chain be? What is known and what is conjectured are at opposite ends of the spectrum. It is unknown whether there are infinitely many Cunningham chains of length 2, but it is conjectured that there are infinitely many Cunningham chains of all lengths.

According to this page, the longest known Cunningham chains of the first kind has length 17, and the longest known Cunningham chain of the second kind has length 19. We can verify these results with the following Python code.

from sympy import isprime

def chain_length(start, kind):
    p = start
    c = 0
    while isprime(p):
        c += 1
        p = 2*p + kind
    return c

print(chain_length(2759832934171386593519, 1))
print(chain_length(79910197721667870187016101, -1))

Bi-twin chains

A number n is the basis of a bi-twin chain of length k if n − 1 is the start of a Cunningham chain of the first kind of length k and n + 1 is the start of a Cunningham chain of the second kind of length k.

I say more about bi-twin prime chains in the next post.

Primecoin

Primecoin was one of the first cryptocurrencies, coming out four years after Bitcoin. Primecoin is still going, though its market cap is six orders magnitude smaller than that of Bitcoin.

What’s interesting about Primecoin is that it uses finding prime chains as its proof of work task [1]. To mint a new Primecoin block, you must find a prime chain of the required length whose origin a multiple of the hash of the block header [2].

Primecoin allows any of the three kinds of prime chains mentioned above: Cunningham chains of the first or second kind, or a bi-twin prime chain. Primecoin adjusts its mining difficulty over time by varying the length of Cunningham or bi-twin chain needed to mint a new block.

There are also cryptocurrencies based on finding prime clusters and prime gaps.

Related posts

[1] Strictly speaking, Primecoin requires finding probable prime chains, as explained here.

[2] The origin of a prime chain is n if the first item in a Cunningham chain of the first kind is n + 1, or if the first item in a Cunningham chain of the first kind or a bi-twin chain is n − 1.

Largest known compositorial prime

Posted on by John

I ran across a blog post here that said a new record has been set for the largest compositorial prime. [1]

OK, so what is a compositorial prime? It is a prime number of the form

n! / n# + 1

where n# denotes n primorial, the product of the prime numbers no greater than n.

The newly discovered prime is

N = 751882!/751882# + 1

It was described in the article cited above as

751882!/751879# + 1,

but the two numbers are the same because there are no primes greater than 751879 and less than 751882, i.e.

751879# = 751882#.

About how large is N? We can calculate the log of the numerator easily enough:

>>> import scipy.special
>>> scipy.special.loggamma(751883)
9421340.780760147

However, the denominator is harder to compute. According to OIES we have

n# = exp((1 + o(1)) n)

which would give us the estimate

log(751882#) ≈ 751882.

So

log10(N) = log(N) / log(10) ≈ (9421340 − 751882) / log(10) ≈ 3765097.

According to this page,

log10(N) = 3765620.3395779

and so our approximation above was good to four figures.

So N has between 3 and 4 million digits, making it much smaller than the largest known prime, which has roughly 41 million digits. Overall, N is the 110th largest known prime.

 

[1] I misread the post at first and thought it said there was a new prime record (skipping over the “compositorial” part) and was surprised because the number is not a Mersenne number. For a long time now the largest known prime has been a Mersenne prime because there is a special algorithm for testing whether Mersenne numbers are prime, one that is much more efficient than testing numbers in general.

 

log2(3) and log2(5)

Posted on by John

AlmostSure on X pointed out that

log2 3 ≈ 19/12,

an approximation that’s pretty good relative to the size of the denominator. To get an approximation that’s as accurate or better requires a larger denominator for log2 5.

log2 5 ≈ 65/28

This above observations are correct, but are they indicative of a more general pattern? Is log2 3 easier to approximate than log2 5 using rational numbers? There are theoretical ways to quantify this—irrationality measures—but they’re hard to compute.

If you look at the series of approximations for both numbers, based on continued fraction convergents, the nth convergent for log2 5 is more accurate than the nth convergent for log2 3, at least for the first 16 terms. After that I ran out of floating point precision and wasn’t sufficiently interested to resort to extended precision.

Admittedly this is a non-standard way to evaluate approximation error. Typically you look at the approximation error relative to the size of the denominator, not relative to the index of the convergents.

Here’s a more conventional comparison, plotting the log of approximation error against the log of the denominators.

Continued fraction posts

Multiples with no large digits

Posted on by John

Here’s a curious theorem I stumbled across recently [1]. Take an integer N which is not a multiple of 10. Then there is some multiple of N which only contains the digits 1, 2, 3, 4, and 5.

For example, my business phone number 8324228646 has a couple 8s and a couple 6s. But

6312 × 8324228646 = 52542531213552

which contains only digits 1 through 5.

For a general base b, let p be the smallest prime factor of b. Then for every integer N that is not a multiple of b, there is some multiple of N whose base b representation contains only the digits 1, 2, 3, …, b/p.

This means that for every number N that is not a multiple of 16, there is some k such that the hex representation of kN contains only the digits 1 through 8. For example, if we take the magic number at the beginning of every Java class file, 0xCAFEBABE, we find

1341 × CAFEBABEhex = 42758583546hex.

In the examples above, we’re looking for multiple containing only half the possible digits. If the largest prime dividing the base is larger than 2 then we can find a multiples with digits in a smaller range. For example, in base 35 we can find a multiple containing only the digits 1 through 7.

[1] Gregory Galperin and Michael Reid. Multiples Without Large Digits. The American Mathematical Monthly, Vol. 126, No. 10 (December 2019), pp. 950-951.