# Least common multiple of the first n positive integers

Here’s a surprising result: The least common multiple of the first n positive integers is approximately exp(n).

More precisely, let φ(n) equal the log of the least common multiple of the numbers 1, 2, …, n. There are theorems that give upper and lower bounds on how far φ(n) can be from n. We won’t prove or even state these bounds here. See [1] for that. Instead, we’ll show empirically that φ(n) is approximately n.

Here’s some Python code to plot φ(n) over n. The ratio jumps up sharply after the first few values of n. In the plot below, we chop off the first 20 values of n.

from scipy import arange, empty
from sympy.core.numbers import ilcm
from sympy import log
import matplotlib.pyplot as plt

N = 5000
x = arange(N)
phi = empty(N)

M = 1
for n in range(1, N):
M = ilcm(n, M)
phi[n] = log(M)

a = 20
plt.plot(x[a:], phi[a:]/x[a:])
plt.xlabel("$n$")
plt.ylabel("$\phi(n) / n$")
plt.show()

Here’s the graph this produces.

[1] J. Barkley Rosser and Lowell Schoenfeld. Approximate formulas for some functions of prime numbers. Illinois Journal of Mathematics, Volume 6, Issue 1 (1962), 64-94. (On Project Euclid)

# Quicksort and prime numbers

The average number of operations needed for quicksort to sort a list of n items is approximately 10 times the nth prime number.

Here’s some data to illustrate this.

|------+-----------------+---------|
|    n | avg. operations | 10*p(n) |
|------+-----------------+---------|
|  100 |          5200.2 |    5410 |
|  200 |         12018.3 |   12230 |
|  300 |         19446.9 |   19870 |
|  400 |         27272.2 |   27410 |
|  500 |         35392.2 |   35710 |
|  600 |         43747.3 |   44090 |
|  700 |         52297.8 |   52790 |
|  800 |         61015.5 |   61330 |
|  900 |         69879.6 |   69970 |
| 1000 |         78873.5 |   79190 |
| 1100 |         87984.4 |   88310 |
| 1200 |         97201.4 |   97330 |
| 1300 |        106515.9 |  106570 |
| 1400 |        115920.2 |  116570 |
| 1500 |        125407.9 |  125530 |
| 1600 |        134973.5 |  134990 |
| 1700 |        144612.1 |  145190 |
| 1800 |        154319.4 |  154010 |
| 1900 |        164091.5 |  163810 |
| 2000 |        173925.1 |  173890 |
|------+-----------------+---------|

The maximum difference between the quicksort and prime columns is about 4%. In the latter half of the table, the maximum error is about 0.4%.

What’s going on here? Why should quicksort be related to prime numbers?!

The real mystery is the prime number theorem, not quicksort. The prime number theorem tells us that the nth prime number is approximately n log n. And the number of operations in an efficient sort is proportional to n log n. The latter is easier to see than the former.

A lot of algorithms have run time proportional to n log n: mergesort, heapsort, FFT (Fast Fourier Transform), etc. All these have run time approximately proportional to the nth prime.

Now for the fine print. What exactly is the average run time for quicksort? It’s easy to say it’s O(n log n), but getting more specific requires making assumptions. I used as the average number of operations 11.67 n log n – 1.74 n based on Knuth’s TAOCP, Volume 3. And why 10 times the nth prime and not 11.67? I chose 10 to make the example work better. For very large values on n, a larger coefficient would work better.

# Leading digits of powers of 2

The first digit of a power of 2 is a 1 more often than any other digit. Powers of 2 begin with 1 about 30% of the time. This is because powers of 2 follow Benford’s law. We’ll prove this below.

When is the first digit of 2n equal to k? When 2n is between k × 10p and (k+1) × 10p for some positive integer p. By taking logarithms base 10 we find that this is equivalent to the fractional part of n log102 being between log10 k and log10 (k + 1).

The map

x ↦ ( x + log10 2 ) mod 1

is ergodic. I wrote about irrational rotations a few weeks ago, and this is essentially the same thing. You could scale x by 2π and think of it as rotations on a circle instead of arithmetic mod 1 on an interval. The important thing is that log10 2 is irrational.

Repeatedly multiplying by 2 is corresponds to adding log10 2 on the log scale. So powers of two correspond to iterates of the map above, starting with x = 0. Birkhoff’s Ergodic Theorem tells us that the number of times iterates of this map fall in the interval [ab] equals b – a. So for k = 1, 2, 3, … 9, the proportion of powers of 2 start with k is equal to  log10 (k + 1) –  log10 (k) =  log10 ((k + 1) / k).

This is Benford’s law. In particular, the proportion of powers of 2 that begin with 1 is equal to  log10 (2) = 0.301.

Note that the only thing special about 2 is that log10 2 is irrational. Powers of 3 follow Benford’s law as well because log10 3 is also irrational. For what values of b do powers of b not follow Benford’s law? Those with log10 b rational, i.e. powers of 10. Obviously powers of 10 don’t follow Benford’s law because their first digit is always 1!

[Interpret the “!” above as factorial or exclamation as you wish.]

Let’s look at powers of 2 empirically to see Benford’s law in practice. Here’s a simple program to look at first digits of powers of 2.

count = [0]*10
N = 10000

def first_digit(n):
return int(str(n)[0])

for i in range(N):
n = first_digit( 2**i )
count[n] += 1

print(count)

Unfortunately this only works for moderate values of N. It ran in under a second with N set to 10,000 but for larger values of N it rapidly becomes impractical.

Here’s a much more efficient version that ran in about 2 seconds with N = 1,000,000.

from math import log10
N = 1000000
count = [0]*10

def first_digit_2_exp_e(e):
r = (log10(2.0)*e) % 1
for i in range(2, 11):
if r < log10(i):
return i-1
for i in range(N):
n = first_digit_2_exp_e( i )
count[n] += 1

print(count)

You could make it more efficient by caching the values of log10 rather than recomputing them. This brought the run time down to about 1.4 seconds. That’s a nice improvement, but nothing like the orders of magnitude improvement from changing algorithms.

Here are the results comparing the actual counts to the predictions of Benford’s law (rounded to the nearest integer).

|---------------+--------+-----------|
| Leading digit | Actual | Predicted |
|---------------+--------+-----------|
|             1 | 301030 |    301030 |
|             2 | 176093 |    176091 |
|             3 | 124937 |    124939 |
|             4 |  96911 |     96910 |
|             5 |  79182 |     79181 |
|             6 |  66947 |     66948 |
|             7 |  57990 |     57992 |
|             8 |  51154 |     51153 |
|             9 |  45756 |     45757 |
|---------------+--------+-----------|

The agreement is almost too good to believe, never off by more than 2.

Are the results correct? The inefficient version relied on integer arithmetic and so would be exact. The efficient version relies on floating point and so it’s conceivable that limits of precision caused a leading digit to be calculated incorrectly, but I doubt that happened. Floating point is precise to about 15 significant figures. We start with log10(2), multiply it by numbers up to 1,000,000 and take the fractional part. The result is good to around 9 significant figures, enough to correctly calculate which log digits the result falls between.

Update: See Andrew Dalke’s Python script in the comments. He shows a way to efficiently use integer arithmetic.

# Uniform distribution of powers mod 1

A few days ago I wrote about how powers of the golden ratio are nearly integers but powers of π are not. This post is similar but takes a little different perspective. Instead of looking at how close powers are to the nearest integers, we’ll look at how close they are to their floor, the largest integer below. Put another way, we’ll throw away the integer parts and look at the decimal parts.

First a theorem:

For almost all x > 1, the sequence (xn) for n = 1, 2, 3, … is u.d. mod 1. [1]

Here “almost all” is a technical term meaning that the set of x‘s for which the statement above does not hold has Lebesgue measure zero. The abbreviation “u.d.” stands for “uniformly distributed.” A sequence uniformly distributed mod 1 if the fractional parts of the sequence are distributed like uniform random variables.

Even though the statement holds for almost all x, it’s hard to prove for particular values of x. And it’s easy to find particular values of x for which the theorem does not hold.

From [1]:

… it is interesting to note that one does not know whether sequences such as (en), (πn), or even ((3/2)n) are u.d. mod 1 or not.

Obviously powers of integers are not u.d. mod 1 because their fractional parts are all 0. And we’ve shown before that powers of the golden ratio and powers of the plastic constant are near integers, i.e. their fractional parts cluster near 0 and 1.

The curious part about the quote above is that it’s not clear whether powers of 3/2 are uniformly distributed mod 1. I wouldn’t expect powers of any rational number to be u.d. mod 1. Either my intuition was wrong, or it’s right but hasn’t been proved, at least not when [1] was written.

The next post will look at powers of 3/2 mod 1 and whether they appear to be uniformly distributed.

* * *

[1] Kuipers and Niederreiter, Uniform Distribution of Sequences

# Plastic powers

Last week I wrote a blog post showing that powers of the golden ratio are nearly integers. Specifically, the distance from φn to the nearest integer decreases exponentially as n increases. Several people pointed out that the golden constant is a Pisot number, the general class of numbers whose powers are exponentially close to integers.

The so-called plastic constant P is another Pisot number, in fact the smallest Pisot number. P is the real root of x3x – 1 = 0.

Because P is a Pisot number, we know that its powers will be close to integers, just like powers of the golden ratio, but the way they approach integers is more interesting. The convergence is slower and less regular.

We will the first few powers of P, first looking at the distance to the nearest integer on a linear scale, then looking at the absolute value of the distance on a logarithmic scale.

As a reminder, here’s what the corresponding plots looked like for the golden ratio.

# Golden powers are nearly integers

This morning I was reading Terry Tao’s overview of the work of Yves Meyer and ran across this line:

The powers φ, φ2, φ3, … of the golden ratio lie unexpectedly close to integers: for instance, φ11 = 199.005… is unusually close to 199.

I’d never heard that before, so I wrote a little code to see just how close golden powers are to integers.

Here’s a plot of the difference between φn and the nearest integer:

(Note that if you want to try this yourself, you need extended precision. Otherwise you’ll get strange numerical artifacts once φn is too large to represent exactly.)

By contrast, if we make the analogous plot replacing φ with π we see that the distance to the nearest integer looks like a uniform random variable:

The distance from powers of φ to the nearest integer decreases so fast that cannot see it in the graph for moderate sized n, which suggests we plot the difference on the log scale. (In fact we plot the log of the absolute value of the difference since the difference could be negative and the log undefined.) Here’s what we get:

After an initial rise, the curve is apparently a straight line on a log scale, i.e. the absolute distance to the nearest integer decreases almost exactly exponentially.

Related posts:

# Inverse Fibonacci numbers

As with the previous post, this post is a spinoff of a blog post by Brian Hayes. He considers the problem of determining whether a number n is a Fibonacci number and links to a paper by Gessel that gives a very simple solution: A positive integer n is a Fibonacci number if and only if either 5n2 – 4 or 5n2 + 4 is a perfect square.

If we know n is a Fibonacci number, how can we tell which one it is? That is, if n = Fm, how can we find m?

For large m, Fm is approximately φm / √ 5 and the error decreases exponentially with m. By taking logs, we can solve for m and round the result to the nearest integer.

We can illustrate this with SymPy. First, let’s get a Fibonacci number.

>>> from sympy import *
>>> F = fibonacci(100)
>>> F
354224848179261915075

Now let’s forget that we know F is a Fibonacci number and test whether it is one.

>>> sqrt(5*F**2 - 4)
sqrt(627376215338105766356982006981782561278121)

Apparently 5F2 – 4 is not a perfect square. Now let’s try 5F2 + 4.

>>> sqrt(5*F**2 + 4)
792070839848372253127

Bingo! Now that we know it’s a Fibonacci number, which one is it?

>>> N((0.5*log(5) + log(F))/log(GoldenRatio), 10)
100.0000000

So F must be the 100th Fibonacci number.

It looks like our approximation gave an exact result, but if we ask for more precision we see that it did not.

>>> N((0.5*log(5) + log(F))/log(GoldenRatio), 50)
99.999999999999999999999999999999999999999996687654

Related posts:

# Infinite primes via Fibonacci numbers

A well-known result about Fibonacci numbers says

gcd(Fm, Fn) = Fgcd(m, n)

In words, the greatest common divisor of the mth and nth Fibonacci numbers is the gth Fibonacci number where g is the greatest common divisor of m and n. You can find a proof here.

M. Wunderlich used this identity to create a short, clever proof that there are infinitely many primes.

Suppose on the contrary that there are only k prime numbers, and consider the set of Fibonacci numbers with prime indices: Fp1, Fp2, … Fpk. The Fibonacci theorem above tells us that these Fibonacci numbers are pairwise relatively prime: each pair has greatest common divisor F1 = 1.

Each of the Fibonacci numbers in our list must have only one prime factor. Why? Because we have assumed there are only k prime numbers, and no two of our Fibonacci numbers share a prime factor. But F19 = 4181 = 37*113. We’ve reached a contradiction, and so there must be infinitely many primes.

Source: M. Wunderlich, Another proof of the infinite primes theorem. American Mathematical Monthly, Vol. 72, No. 3 (March 1965), p. 305.

Here are a couple more unusual proofs that there are infinitely many primes. The first uses a product of sines. The second is from Paul Erdős. It also gives a lower bound on π(N), the number of primes less than N.

# Three proofs that 2017 is prime

Aaron Meurer asked on Twitter whether there’s a proof that 2017 is prime that would fit into 140 characters.

sqrt(2017) < 45.
2017 not divisible by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, or 43.
Ergo prime.

I’m not sure that’s what he had in mind. There’s some implied calculation (which I didn’t actually do), so it’s kinda cheating. It would be interesting if there were something special about 2017 that would allow a more transparent proof.

(Implicit in the proof above is the theorem that if a number has a prime factor, it has a prime factor less than it’s square root. If you multiply together numbers bigger than the square root of n, you get a product bigger than n.)

Then for fun I gave two more proofs that are more sophisticated but would require far too much work to actually carry out by hand.

The first uses Fermat’s little theorem:

For 0 < a < 2017, a2016 – 1 is divisible by 2017.
2017 is not one of the three Carmichael numbers < 2465.
Ergo prime.

Fermat’s little theorem says that if p is prime, then for any 0 < ap, ap – 1 – 1 is divisible by p. This is actually an efficient way to prove that a number is not prime. Find a number a such that the result doesn’t hold, and you’ve proved that p isn’t prime. For small numbers, the easiest way to show a number is not prime is to show its factors. But for very large numbers, such as those used in cryptography, it’s efficient to have a way to prove that a number has factors without having to actually produce those factors.

Unfortunately, Fermat’s little theorem gives a necessary condition for a number to be prime, but not a sufficient condition. It can appear to be prime for every witness (the bases a are called witnesses) and still not be a prime. The Carmichael numbers pass the Fermat primailty test without being prime. The first four are 561, 1105, 1729, and 2465.

For more on using Fermat’s little theorem to test for primality, see Probability that a number is prime.

2016! + 1 is divisible by 2017, and so by Wilson’s theorem 2017 is prime.

Unlike Fermat’s little theorem, Wilson’s theorem gives necessary and sufficient conditions for a number to be prime. A number n is prime if and only if (n-1)! + 1 is divisible by n. In theory you could use Wilson’s theorem to test whether a number is prime, but this would be horrendously inefficient. 2016! has 5,789 digits. (You can find out how many digits n! has without computing it using a trick described here.)

Despite its inefficiency, you can actually use Wilson’s theorem and SymPy to prove that 2017 is prime.

>>> from sympy import factorial
>>> x = factorial(2016) + 1
>>> x % 2017
0

# A short, unusual proof that there are infinitely many primes

Sam Northshield [1] came up with the following clever proof that there are infinitely many primes.

Suppose there are only finitely many primes and let P be their product. Then

The original publication gives the calculation above with no explanation. Here’s a little commentary to explain the calculation.

Since prime numbers are greater than 1, sin(π/p) is positive for every prime. And a finite product of positive terms is positive. (An infinite product of positive terms could converge to zero.)

Since p is a factor of P, the arguments of sine in the second product differ from those in the first product by an integer multiple of 2π, so the corresponding terms in the two products are the same.

There must be some p that divides 1 + 2P, and that value of p contributes the sine of an integer multiple of π to the product, i.e. a zero. Since one of the terms in the product is zero, the product is zero. And since zero is not greater than zero, we have a contradiction.

* * *

[1] A One-Line Proof of the Inﬁnitude of Primes, The American Mathematical Monthly, Vol. 122, No. 5 (May 2015), p. 466

# Computing discrete logarithms with baby-step giant-step algorithm

At first “discrete logarithm” sounds like a contradiction in terms. Logarithms aren’t discrete, not as we usually think of them. But you can define and compute logarithms in modular arithmetic.

What is a logarithm? It’s the solution to an exponential equation. For example, the logarithm base 10 of 2 is the solution to the equation 10x = 2, and so x =0.30103. Similarly, you could look for the logarithm base 10 of 2 modulo 19. This is an integer value of x such that 10x = 2 mod 19, and you can show that 17 is a solution.

If we work modulo an integer n, the discrete logarithm base a of a number y is an integer x such that ax = y. For some values of a there may not be a solution, but there will be a solution if a is a generator of the integers mod n.

Brute force the simplest algorithm for finding discrete logarithms: try x = 1, 2, 3, …, n until you find a value of x satisfying ax = y. The problem with brute force is that the expected run time is on the order of n, and n is often very large in application.

Discrete logarithms are expensive to compute, but we can do better than brute force. (Cryptographic applications take advantage of the fact that discrete logarithms are hard to compute.) There’s a simple algorithm by Daniel Shanks, known as the baby-step giant-step algorithm, that reduces the run time from order n to order roughly √n. (Actually O(√n log n) for reasons we’ll see soon.)

Let s be the ceiling of the square root of n. The idea is to search for x by taking giant steps forward (multiples of s) and baby (single) steps backward.

Let G be the set of pairs (ags mod n, gs) where g ranges from 1 to s. These are the giant steps.

Let B be the set of pairs (yab mod n, b) where b ranges from 0 to s-1. These are the baby steps.

Now look for a pair in G and a pair in B with the matching first components, i.e. yab = ags. Then x = gsb is the required solution.

Constructing the sets G and requires O(s) operations, and matching the two sets takes O(s log s) operations.

Here’s an example, going back to the problem above of finding the discrete log base 10 of 2 mod 19, using a little Python code to help with the calculations.

The square root of 19 is between 4 and 5 so s = 5.

>>> [(2*10**r % 19, r) for r in range(5)]
[(2, 0), (1, 1), (10, 2), (5, 3), (12, 4)]
>>> [(10**(4*t) % 19, 4*t) for t in range(1,6)]
[(6, 4), (17, 8), (7, 12), (4, 16), (5, 20)]

The number 5 appears as the first element of a pair in both B and G. We have (5, 3) in B and (5, 20) in G so x = 20 – 3 = 17.

Related: Applied number theory

# Periods of fractions

Suppose you have a fraction a/b where 0 < ab, and a and b are relatively prime integers. The decimal expansion of a/b either terminates or it has an initial non-repeating part followed by a repeating part.

How long is the non-repeating part? How long is the period of the repeating part?

The answer depends on the prime factorization of the denominator b. If b has the form

b = 2α 5β

then the decimal expansion has r digits where r = max(α, β).

Otherwise b has the factorization

b = 2α 5β d

where d is some integer greater than 1 and relatively prime to 10. Then the decimal expansion of a/b has r non-repeating digits and a repeating part with period s where r is as above, and s is the smallest positive integer such that

d | 10s– 1,

i.e. the smallest s such that 10s – 1 is divisible by d. How do we know there exists any such integer s? This isn’t obvious.

Fermat’s little theorem tells us that

d | 10d – 1 – 1

and so we could take s = d – 1, though this may not be the smallest such s. So not only does s exist, we know that it is at most d – 1. This means that the period of the repeating part of a/b is no more than d – 1 where d is what’s left after factoring out as many 2’s and 5’s from b as possible.

By the way, this means that you can take any integer d, not divisible by 2 or 5, and find some integer k such that dk consists only of 9’s.

Related post: Cyclic fractions

# Münchausen numbers

Baron Münchausen

The number 3435 has the following curious property:

3435 = 33 + 44 + 33 + 55.

It is called a Münchausen number, an allusion to fictional Baron Münchausen. When each digit is raised to its own power and summed, you get the original number back. The only other Münchausen number is 1.

At least in base 10. You could look at Münchausen numbers in other bases. If you write out a number n in base b, raise each of its “digits” to its own power, take the sum, and get n back, you have a Münchausen number in base b. For example 28 is a Münchausen number in base 9 because

28ten = 31nine = 33 + 11

Daan van Berkel proved that there are only finitely many Münchausen in any given base. In fact, he shows that a Münchausen number in base b cannot be greater than 2bb, and so you could do a brute-force search to find all the Münchausen numbers in any base.

The upper bound 2bb grows very quickly with b and so brute force becomes impractical for large b. If you wanted to find all the hexadecimal Münchausen numbers you’d have to search 2*1616 = 36,893,488,147,419,103,232 numbers. How could you do this more efficiently?

# Prime factors, phone numbers, and the normal distribution

Telephone numbers typically have around three distinct prime factors.

The length of a phone number varies by country, but US a phone number is a 10 digit number, and 10-digit numbers are often divisible by three different prime numbers, give or take a couple. Assuming phone numbers are scattered among possible 10-digit numbers in a way that doesn’t bias their number of prime factors, these numbers will often have between 1 and 5 prime factors. If a country has 9- or 11-digit phone numbers, the result is essentially the same.

Let ω(n) be the number of distinct prime factors of n. Then the Erdős–Kac theorem says roughly that ω(n) is distributed like a normal random variable with mean and variance log log n. More precisely, fix two numbers a and b.  For a given value of x, count the proportion of positive integers less than x where

(ω(n) – log log n) / sqrt( log log n)

is between a and b. Then in the limit as x goes to infinity, this proportion approaches the probability that a standard normal random variable is between a and b.

So by that heuristic, the number of distinct prime factors of a 10-digit number is approximately normally distributed with mean and variance log log 10^11 = 3.232, and such a distribution is between 1 and 5 around 73% of the time.

My business phone number, for example, is 8324228646. Obviously this is divisible by 2. In fact it equals 2 × 32 × 462457147, and so it has exactly three distinct prime factors: 2, 3, and 462457147.

Here’s how you could play with this using Python.

from sympy.ntheory import factorint

def omega(n):
return len(factorint(n))

I looked in SymPy and didn’t see an implementation of ω(n) directly, but it does have a function factorint that returns the prime factors of a number, along with their multiplicities, in a dictionary. So ω(n) is just the size of that dictionary.

I took the first 20 phone numbers in my contacts and ran them through omega and got results consistent with what you’d expect from the theory above. One was prime, and none had more than five factors.

# Benford’s law, chi-square, and factorials

A while back I wrote about how the leading digits of factorials follow Benford’s law. That is, if you look at just the first digit of a sequence of factorials, they are not evenly distributed. Instead, 1’s are most popular, then 2’s, etc. Specifically, the proportion of factorials starting with n is roughly log10(1 + 1/n).

Someone has proved that the limiting distribution of leading digits of factorials exactly satisfies Benford’s law. But if we didn’t know this, we might use a chi-square statistic to measure how well the empirical results match expectations. As I argued in the previous post, statistical tests don’t apply here, but they can be useful anyway in giving us a way to measure the size of the deviations from theory.

Benford’s law makes a better illustration of the chi-square test than the example of prime remainders because the bins are unevenly sized, which they’re allowed to be in general. In the prime remainder post, they were all the same size.

The original post on leading digits of factorial explains why we compute the leading digits the way we do. Only one detail has changed: the original post used Python 2 and this one uses Python 3. Integer division was the default in Python 2, but now in Python 3 we have to use // to explicitly ask for integer division, floating point division being the new default.

Here’s a plot of the distribution of the leading digits for the first 500 factorials.

And here’s code to compute the chi-square statistic:

from math import factorial, log10

while n > 9:
n = n//10
return n

def chisq_stat(O, E):
return sum( [(o - e)**2/e for (o, e) in zip(O, E)] )

# Waste the 0th slot to make the code simpler.
digits = [0]*10

N = 500
for i in range(N):
digits[ leading_digit_int( factorial(i) ) ] += 1

expected = [ N*log10(1 + 1/n) for n in range(1, 10) ]

print( chisq_stat(digits[1:], expected) )

This gives a chi-square statistic of 7.693, very near the mean value of 8 for a chi-square distribution with eight degrees of freedom. (There are eight degrees of freedom, not nine, because if we know how many factorials start with the digits 1 through 8, we know how many start with 9.)

So the chi-square statistic suggests that the deviation from Benford’s law is just what we’d expect from random data following Benford’s law. And as we said before, this suggestion turns out to be correct.

Related posts: