# Prime numbers

## Project Euler 146: Investigating a Prime Pattern

In Problem 146 of Project Euler we are working with primes again, and some quite big ones even. The problem reads

The smallest positive integer n for which the numbers n2+1, n2+3, n2+7, n2+9, n2+13, and n2+27 are consecutive primes is 10. The sum of all such integers n below one-million is 1242490.

What is the sum of all such integers n below 150 million?

At first I thought I could just make a sieve up to 150 million and then check if the numbers were contained in that. However, rereading the problem I realized I was completely wrong. So in a pure brute force solution we would need to check 150 million values of n and up to 13 numbers for each, since we both need to check that the given numbers are prime. But also that the odd numbers inbetween are not prime. So potentially we have to check 1950 million numbers for primality, which is a moderately expensive operation. Continue reading →

Posted by Kristian in Project Euler, 8 comments

## Project Euler 134: Prime pair connection

With a little help from Wikipedia I found Problem 134 of Project Euler rather easy to solve.  Well it was easy after solving some issues I had with integer overflows and several other bugs. But anyway, the problem reads

Consider the consecutive primes p1 = 19 and p2 = 23. It can be verified that 1219 is the smallest number such that the last digits are formed by p1 whilst also being divisible by p2.

In fact, with the exception of p1 = 3 and p2 = 5, for every pair of consecutive primes, p2 > p1, there exist values of n for which the last digits are formed by p1 and n is divisible by p2. Let S be the smallest of these values of n.

Find ΣS for every pair of consecutive primes with 5 ≤ p1 ≤ 1000000.

Posted by Kristian in Project Euler, 5 comments

## Project Euler 133: Repunit nonfactors

Problem 133 of Project Euler is a continuation of Problem 132 and Problem 129 in which we are supposed to find the some prime numbers which are not factors of R(10n) for any n. In fact the problem reads

A number consisting entirely of ones is called a repunit. We shall define R(k) to be a repunit of length k; for example, R(6) = 111111.

Let us consider repunits of the form R(10n).

Although R(10), R(100), or R(1000) are not divisible by 17, R(10000) is divisible by 17. Yet there is no value of n for which R(10n) will divide by 19. In fact, it is remarkable that 11, 17, 41, and 73 are the only four primes below one-hundred that can be a factor of R(10n).

Find the sum of all the primes below one-hundred thousand that will never be a factor of R(10n).

Posted by Kristian in Project Euler, 6 comments

## Project Euler 132: Large repunit factors

In problem 132 of Project Euler we are going back to working with repunits in a problem that reads

A number consisting entirely of ones is called a repunit. We shall define R(k) to be a repunit of length k.

For example, R(10) = 1111111111 = 11 x 41 x 271 x 9091, and the sum of these prime factors is 9414.

Find the sum of the first forty prime factors of R(109).

Posted by Kristian in Project Euler, 4 comments

## Project Euler 131: Determining primes, p, for which n^3 + n^2p is a perfect cube.

I am pretty sure the Problem 131 of Project Euler can be brute forced. However, if you start digging you will see that there is a really beautiful solution to the problem that reads

There are some prime values, p, for which there exists a positive integer, n, such that the expression n3 + n2p is a perfect cube.

For example, when p = 19, 83 + 82x19 = 123.

What is perhaps most surprising is that for each prime with this property the value of n is unique, and there are only four such primes below one-hundred.

How many primes below one million have this remarkable property?

Posted by Kristian in Project Euler, 4 comments

## Project Euler 128: Which tiles in the hexagonal arrangement have prime differences with neighbours?

Problem 128 is a really interesting problem about prime numbers. It reads

A hexagonal tile with number 1 is surrounded by a ring of six hexagonal tiles, starting at “12 o’clock” and numbering the tiles 2 to 7 in an anti-clockwise direction.

New rings are added in the same fashion, with the next rings being numbered 8 to 19, 20 to 37, 38 to 61, and so on. The diagram below shows the first three rings.

By finding the difference between tile n and each its six neighbours we shall define PD(n) to be the number of those differences which are prime.

For example, working clockwise around tile 8 the differences are 12, 29, 11, 6, 1, and 13. So PD(8) = 3.

In the same way, the differences around tile 17 are 1, 17, 16, 1, 11, and 10, hence PD(17) = 2.

It can be shown that the maximum value of PD(n) is 3.

If all of the tiles for which PD(n) = 3 are listed in ascending order to form a sequence, the 10th tile would be 271.

Find the 2000th tile in this sequence.

Posted by Kristian in Project Euler, 2 comments

## Project Euler 123: Determining the remainder when (pn − 1)^n + (pn + 1)^n is divided by pn^2

Problem 123 of Project Euler reads

Let pn be the nth prime: 2, 3, 5, 7, 11, …, and let r be the remainder when (pn-1)n + (pn+1)n is divided by pn2.

For example, when n = 3, p3 = 5, and 43 + 63 = 280 ≡ 5 mod 25.

The least value of n for which the remainder first exceeds 109 is 7037.

Find the least value of n for which the remainder first exceeds 1010.

Posted by Kristian in Project Euler, 7 comments

## News on the ABC conjecture

Here is some news of the possible breakthrough of the ABC conjecture. It  might already be commonly known, but  it is something I only recently discovered was going on. There are rumors that Shinichi Mochizuki from Kyoto university has solved the abc conjecture. You can find his proof here. Not that I understood anything of it, but I though a link would be appropriate. But don’t despair yet, I will have a few more links in this regard. Continue reading →

Posted by Kristian in Math, 1 comment

## Pandigitial primes

I got a fun little question from Jean-Marie by email the other day. With the disclaimer that I don’t really have the time to solve that many puzzles and therefore probably wont be able to give you a solution to puzzles like this one. This one triggered my curiosity and I managed to find a rather cure solution.

Therefore I will post it to you all and let you wonder a bit about it. If you find the solution feel free to post it in the comments.

The question is simple:

Find all the primes that contain 9 different digits (0 excluded).

Good luck 🙂

Posted by Kristian in Math, 3 comments

## Project Euler 111: Search for 10-digit primes containing the maximum number of repeated digits.

So in Problem 111 of Project Euler we are back at dealing with prime numbers. And rather large ones of them. But before going into details, here is the problem description

Considering 4-digit primes containing repeated digits it is clear that they cannot all be the same: 1111 is divisible by 11, 2222 is divisible by 22, and so on. But there are nine 4-digit primes containing three ones:

1117, 1151, 1171, 1181, 1511, 1811, 2111, 4111, 8111

We shall say that M(nd) represents the maximum number of repeated digits for an n-digit prime where d is the repeated digit, N(nd) represents the number of such primes, and S(nd) represents the sum of these primes.

So M(4, 1) = 3 is the maximum number of repeated digits for a 4-digit prime where one is the repeated digit, there are N(4, 1) = 9 such primes, and the sum of these primes is S(4, 1) = 22275. It turns out that for d = 0, it is only possible to have M(4, 0) = 2 repeated digits, but there are N(4, 0) = 13 such cases.

In the same way we obtain the following results for 4-digit primes.

 Digit, d M(4, d) N(4, d) S(4, d) 0 2 13 67061 1 3 9 22275 2 3 1 2221 3 3 12 46214 4 3 2 8888 5 3 1 5557 6 3 1 6661 7 3 9 57863 8 3 1 8887 9 3 7 48073

For d = 0 to 9, the sum of all S(4, d) is 273700.

Find the sum of all S(10, d).

Posted by Kristian in Project Euler, 8 comments