Primes of the form $4k+1$

Abstract

The main goal of this chapter is to prove that there are infinitely many primes of the form $4k+1$. We will also state the Law of Quadratic Reciprocity.

Appendices

Exercises

• 6.1 Suppose we have a non-constant polynomial $f(x) \in \mathbb Z[x]$. Show that the set of prime numbers p such that $p \mid f(n)$ for some n is infinite.

• 6.2 Show for every non-constant polynomial $f(x) \in \mathbb Z[x]$ there are infinitely many values of n for which f(n) is not prime.

• 6.3 Show that there are infinitely many primes of the form

  • a. $8k+1$;

  • b. $8k+3$;

  • c. $5k+4$;

  • d. $12k+1$;

  • e. $12k+5$;

  • f. $12k+7$;

  • g $12k+11$.

• 6.4 Compute the following Legendre symbols:

  • a. (13 / 29);

  • b (67 / 193);

  • c. (30 / 103);

  • d. (62 / 569).

• 6.5 Give a group-theoretic interpretation for the Legendre symbol.

• 6.6 Suppose p is an odd prime, and $p \not \mid a$. Show that the congruence $ax^2 + bx + c \equiv 0 \ \mathrm {mod}\ p$ is solvable if and only if $u^2 \equiv b^2 - 4 ac \ \mathrm {mod}\ p$ is solvable.

• 6.7 Give a characterization for all primes p for which the equation $x^2 + 2x + 3 \equiv 0 \ \mathrm {mod}\ p$ is solvable.

• 6.8 Determine all primes p that satisfy $(7/p)=+1$.

• 6.9 Prove that a prime p is of the form $x^2 - 2y^2$ if and only if $p=2$ or $p \equiv \pm 1 \ \mathrm {mod}\ 8$.

• 6.10 Prove if $(n/p)=-1$, then

  $$ \sum _{d \mid n} d^{\frac{p-1}{2}} \equiv 0 \mod p. $$

• 6.11 Determine the product of all quadratic residues modulo p.

• 6.12 Verify the identity

  $$ x^8 - 16 = (x^2 - 2)(x^2 + 2)((x-1)^2+1)((x+1)^2+1). $$

  Use the identity to determine the number of solutions of

  $$ x^8 \equiv 16 \mod p. $$

• 6.13 Determine the number of solutions of the congruence

  $$ x^6 - 11 x^4 + 36 x^2 -36 \equiv 0 \mod p. $$

• 6.14 Show that if $p \mid n^4 - n^2 +1$ for some $n \in \mathbb Z$, then $p \equiv 1 \ \mathrm {mod}\ 1 2$.

• 6.15 Compute $\sum _{r=1}^{p-2} (r(r+1)/p)$.

• 6.16 Let $p>2$ be prime. Determine the number of $1 \le n \le p-2$ such that n and $n+1$ are both quadratic residues modulo p. To do this, consider

  $$ \frac{1}{4}\sum _{n=1}^{p-2} \left( 1 + \left( \frac{n}{p}\right) \right) \left( 1 + \left( \frac{n+1}{p}\right) \right) . $$

• 6.17 Show that if n is not a perfect square, there are infinitely many primes p such that $(n/p)=-1$.

• 6.18 ($\maltese$) We saw in Exercise 2.60 that $p=2^{17}-1$ is prime. Compute the quadratic residue symbols (q / p) for q every prime less than 20.

• 6.19 Prove that there are arbitrarily long non-constant arithmetic progressions such that every two terms of the arithmetic progression are relatively prime.

• 6.20 Let $k \in \mathbb N$. Show that there are integers a, b such that for all $j \in \mathbb N$ the number of divisors of $a + bj$ is divisible by k.

• 6.21 Fix a natural number l. Assuming Theorem 5.11 prove every arithmetic progression $a + bk$, $k \ge 0$, with $\gcd (a, b) = 1$, contains infinitely many terms which are products of l distinct primes.

• 6.22 The goal of this exercise is to show that if $n \in \mathbb N$, then there are infinitely many primes of the form $nk+1$.

  • a. Show that for each $d \in \mathbb N$ there is a monic polynomial $\varPhi _d(x) \in \mathbb Z[x]$, called the d-th cyclotomic polynomial, such that

    $$ \prod _{d \mid n} \varPhi _d(x) = x^n-1. $$

  • b. Show that $\varPhi _1(0) = -1$ and for $d>1$, $\varPhi _d(0)=1$.

  • c. ($\maltese$) Find the first 100 or so cyclotomic polynomials. Pay close attention to the coefficients of the polynomials.

  • d. Suppose $n>1$ and $a \in \mathbb Z$, and let p be a prime divisor of $\varPhi _n(a)$. Then show that $\gcd (a, p)=1$, and if $h = o_p(a)$, $h \mid n$. Furthermore:

    • if $h < n$, then

      $$ a^n-1 \equiv (a+p)^n-1 \equiv 0 \mod p^2; $$

    • if $h < n$, then $p \mid n$;

    • if $p \not \mid n$, then $h=n$ and $p \equiv 1 \ \mathrm {mod}\ n$.

  • e. Conclude there are infinitely many primes of the form $nk+1$.

Notes

1.1 Infinitude of Prime Numbers in The Elements

To get a feel for Euclid’s style of writing, let us state Euclid’s First Theorem, Lemma 2.18:

Theorem 6.10

(Elements, Book VII, Proposition 30). If two numbers by multiplying one another make some number, and any prime number measures [divides] the product, it will also measure one of the original numbers.

It may sound like a historical absurdity that Euclid never stated Theorem 2.19—in fact, this particular fact had to wait almost 2000 years to be put in writing by Gauss. However, any rigorous proof of Theorem 2.19 uses mathematical induction which as a tool was not available to Euclid. At any rate, Euclid used this theorem to prove the irrationality of $\sqrt{n}$ for n non-square, which may have been his original goal in writing the number theoretic parts of The Elements.

This is Euclid’s original formulation of Theorem 6.2:

Theorem 6.11

(Elements, Book IX, Proposition 20). Prime numbers are more than any assigned multitude of prime numbers.

Here we will reproduce Euclid’s original argument. Note that here Euclid illustrates the idea by working out the proof for a special case:

Let A, B, and C be the assigned prime numbers. I say that there are more prime numbers than A, B, and C.

Take the least number DE measured by A, B, and C. Add the unit DF to DE. Then EF is either prime or not.

First, let it be prime. Then the prime numbers A, B, C, and EF have been found which are more than A, B, and C.

Next, let EF not be prime. Therefore it is measured by some prime number. Let it be measured by the prime number G. I say that G is not the same with any of the numbers A, B, and C. If possible, let it be so.

Now A, B, and C measure DE; therefore G also measures DE. But it also measures EF. Therefore G, being a number, measures the remainder, the unit DF, which is absurd.

Therefore G is not the same with any one of the numbers A, B, and C. And by hypothesis it is prime. Therefore the prime numbers A, B, C, and G have been found which are more than the assigned multitude of A, B, and C.

Therefore, prime numbers are more than any assigned multitude of prime numbers.

At the time of this writing, the largest known prime number is $2^{77,232,917}-1$ discovered in 2017. This number has 23, 249, 425 digits. For comparison, the number of atoms in the entire observable universe is a number which is supposed to have about 80 digits. The discovery of this largest prime was part of The Great Internet Mersenne Prime Search accessible through

https://www.mersenne.org/

1.2 Primality testing

The first primality test is due to Eratosthenes (276–194 BCE) who observed that a number n is prime if and only if it is not divisible by any primes up to $\sqrt{n}$; see Exercise 2.18. For n reasonably small this provides a quick way of determining the primality of a number n, but as n gets large this method becomes impractical fairly quickly. Ideally one would like to be able to find a way to tell the primality of a number n in a number of steps that grows like a polynomial in the number of digits of n, and Eratosthenes’ algorithm fails this expectation fairly miserably. Such an algorithm was not available until 2004 when the now-famous paper by M. Agrawal, N. Kayal, and N. Saxena [58] came out.

The algorithm presented in this paper is known as the AKS algorithm. Before AKS what was available in literature was an array of probabilistic algorithms, and some of these work quite well. A favorite example is the Miller–Rabin test [53, §6.3] which is based on Fermat’s Little Theorem in elementary number theory. The Miller–Rabin test is extremely quick, but the trouble is that it gives false positives, in that some composite numbers are marked as primes.

A closely related problem we currently do not know how to solve, which is mentioned in the Notes of Chapter 2, is to factorize a large number as a product of its prime factors with reasonable efficiency. The solution of this problem would have far reaching consequences in terms of cryptography and internet security.

1.3 Twin Prime Conjecture

The following conjecture is considered very difficult:

Conjecture 6.12

(de Polignac, 1849). For every even natural number h, there are infinitely many prime numbers p such that $p+h$ is prime.

The case $h=2$ is known as the Twin Prime Conjecture which at the time of this writing is still open. In 1915 Viggo Brun attempted to prove the Twin Prime Conjecture by proving that

$$\begin{aligned} \sum _{p, p+2 \text { prime}} \frac{1}{p} \end{aligned}$$

(6.5)

diverges. This idea goes back to Euler who proved the infinitude of prime numbers by showing that the series

$$ \sum _{p \text { prime}} \frac{1}{p} $$

diverges. However, surprisingly, Brun proved that the series (6.5) is convergent! Even more surprisingly, the proof was fairly elementary; see Exercise 9.2.7 of [35] and the exercises leading up to it for a presentation of the argument. The theory of sieves that Brun used in his proof has now become a powerful tool in number theory. The next major breakthrough, again involving the theory of sieves, was achieved in 1973 by Jingrun Chen [65] who showed that there are infinitely many primes p such that $p+2$ is the product of at most two primes. In the same paper Chen also proved an approximation to Goldbach’s conjecture; Chen proved every even number is the sum of a prime and a product of at most two primes. In 2005, Goldston, Pintz, and Yıldırım [76] proved a truly remarkable theorem. To state their theorem we will define a piece of notation. For a prime number p, let $p_{\text {next}}$ be the smallest prime number larger than p. Using this notation, the Twin Prime Conjecture would assert the existence of infinitely many primes p such that $p_{\text {next}} - p = 2$. Goldston, Pintz, and Yıldırım used the theory of sieves in an ingenious way to prove

$$ \liminf _{p \rightarrow \infty } \frac{p_{\text {next}} - p}{\log p} =0. $$

It is clear that de Polignac’s conjecture for any h would imply this result, but knowing this result would not give any information about de Polignac’s conjecture. The spectacular work of Yitang Zhang in 2013, building on the techniques of Goldston, Pintz, and Yıldırım, changed the landscape overnight. Zhang [112] showed that there are infinitely many primes p such that

$$ p_{\text {next}} - p < 7 \times 10^7. $$

This was a major achievement in that it showed the difference between consecutive primes was bounded by a uniform bound. In the last few years the bound of $7 \times 10^7$ has been substantially improved by Maynard [85] and the Polymath Project [91]. At the time of this writing we know by [91] that there are infinitely many primes p such that

$$ p_{\text {next}} - p \le 246. $$

At this time it is not clear how to reduce the bound 246, and this might require a new idea. The same paper proves that there are infinitely many primes p such that

$$ (p_{\text {next}})_{\text {next}} - p \le 38130. $$

It would also be of great interest to improve this bound, but, again, this might require an entirely new idea.