10 Interesting Facts About Prime Numbers

by Carson
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Prime numbers

Primes are numbers that are only divisible by 1 and themselves. And these integers have intriguing specialties that are worth mentioning and/or proving yourself. Here are 10 interesting facts about them. Let’s get started.

Table of Contents

  1. Prime numbers are the roots of all other integers (except 1).
  2. If n is prime, 2^n-1 and/or Fn are not always prime.
  3. All primes greater than 7 are always one less and/or one more than an abundant number.
  4. It is conjectured that there are infinitely many twin primes, but this is not proven.
  5. Prime numbers are used in encryption.
  6. Each Mersenne prime is associated with exactly one perfect number.
  7. Prime numbers get rarer.
  8. There are infinitely many prime numbers.
  9. Primality tests are not always accurate.
  10. All primes and prime powers are solitary.

1. Prime Numbers Are the Roots of All Other Integers (Except 1)

Every integer is either a prime number or is a product of a unique set of prime numbers. For example, 910 = 2 * 5 * 7 * 13. These primes sound familiar, right? Guess what! Selecting these four primes without duplicates always results in 910. This is the prime factorization of the integer, and no other integer shares this combination.

This is also one of the reasons why 1 isn’t considered prime. If it is, the rule will not be cut-and-dried, and we will have exceptions instead of applying this to all primes. As 1 only has one factor, it isn’t composite either.

2. If n is Prime, 2n – 1 and/or Fn are not Always Prime

Think of this: 22 – 1 = 3, 23 – 1 = 7, 25 – 1 = 31, and 27 – 1 = 127 are all prime, but does this sequence continue? No, it doesn’t. 211 – 1 = 2047, which is 23 * 89. In fact, Mersenne primes (primes in the form of 2n – 1) are so rare that only 51 of them are found so far.

Prime factorization
Image edited using Canva

F3 (2), F5 (5), F7 (13), F11 (89), and F13 (233) are also primes. But Fibonacci primes are also so rare. The first one to break the sequence is F2 (1). The second is F19 (4181), which is 37 * 113. Coincidentally, there are also only 51 Fibonacci primes known to humanity.

Image edited using Canva

3. All Primes Greater than 7 Are Always One Less and/or One More than an Abundant Number

Abundant numbers are integers whose sum of divisors is greater than twice itself. All multiples of abundant numbers and perfect numbers (except for the perfect number itself) are abundant. Since 6 is perfect, all other multiples of 6 are abundant.

To be a prime, its GCD with any integer except for its multiples must be 1. We’ll focus on 6 as this is what this fact is about. All integers with remainders 1 or 5, when divided by 6, are not divisible by 2 or 3. That means starting from 11, there is always an abundant number (or two) surrounding any prime.

4. It is Conjectured that There Are Infinitely Many Twin Primes, But this is not Proven

What is a twin prime? It is a prime (p) where p – 2 or p + 2 is also prime. The first terms are easy to find: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), etc. However, the average prime gap increases as numbers go up, and pairs of twin primes become hard to find.

Are there infinitely many twin primes, or is there an upper limit? Evidence is leaning toward the former, as it is proven that there are infinitely many prime gaps less than 246. However, the conjecture remained unproven, one of the simplest math problems whose answer is unknown to mankind.

5. Prime Numbers are Used in Encryption

Prime numbers are not just for mathematical purposes. They also contribute to the world as we know it by making asymmetric encryption possible.

An encryption system consists of a private key and a public key. While the private key must be kept secret, the public key can be shared. Because information should be regenerated once received, the algorithms really should be the same, which is easily exploitable.

Therefore, RSA encryption use the product of two enormous prime numbers as part of the public key. This means that if someone eavesdrops and intercepts the information in the middle of a connection, all they receive is garbage unless they can reconstruct the private key, which is incredibly time-consuming and challenging to execute because they need to factor a huge semiprime.

6. Each Mersenne Prime is Associated with Exactly One Perfect Number

The only form of perfect number we know to this day is:

\[2^p-1 * 2^{(p-1)}\]

If 2p – 1 is prime (which indicates that p is also prime).

Prime numbers exactly one less than a power of two are Mersenne primes, and each Mersenne prime yields a perfect number. That means proving that there are infinitely many perfect numbers of this form will prove that there are infinitely many Mersenne primes, and vice versa.

Why does this work? You can find the explanation here.

7. Prime Numbers Get Rarer

There are 4 prime numbers from 1 to 10, while there are only 25 primes from 1 to 100, 168 primes from 1 to 1000, 1229 primes from 1 to 10000, and 9592 primes between 1 and 100000. As you would observe from this pattern, prime numbers are getting rarer, and average prime gaps are becoming greater. But why can this happen?

This phenomenon is because prime numbers accumulate. On average, there are 4.73 factors per integer from 1 to 100, 7.053 factors per integer from 1 to 1000, 9.3643 factors per integer from 1 to 10000, and 11.66714 factors per integer from 1 to 100000. Moreover, if you look up a random integer from 1 to 100000, you will encounter one with more than 11 factors 38.033% of the time. This is compared to only 20.3% when you only consider integers from 1 to 1000.

However, this piece of evidence is not enough. Why aren’t there so many numbers with insane amounts of divisors and so many prime quadruplets? Well, more prime numbers cover more integers. But first, you’ll have to understand what “relatively prime” is. If two integers are relatively prime, their GCD is precisely 1.

Although 50% of numbers are relatively prime with 2, this isn’t the case when it is multiplied by other primes. For instance, only ~33.3% of numbers are relatively prime with 6 (2 * 3), ~26.67% of numbers are relatively prime with 30 (2 * 3 * 5), and ~22.86% of integers are relatively prime with 210. This percentage further decreases if you multiply larger prime numbers. However, this never reaches zero because……

8. There Are Infinitely Many Prime Numbers

There is no upper limit of prime numbers. Even though almost all numbers are composite (with about 71.15% of integers from 1 to 10000000 having 8 factors or more), prime numbers still pop up occasionally, and the largest prime number doesn’t exist because of a simple reason.

Remember that n and n+1 must be relatively prime, right? What if n is the product of the first k primes? If you add or subtract one from the result, you will not obtain a prime factor within the first k primes. For instance, 2 * 3 * 5 * 7 * 11 + 1 = 2311, which is prime. 2 * 3 * 5 * 7 – 1 = 209, which is composite, but is not divisible by any of the primes below 10 (its smallest prime factor is 11).

This pattern goes on until infinity, which means that for every prime number, there are always primes greater than itself. It’s like the fact that adding one to an integer is possible in all cases, no matter how large the number is.

9. Primality Tests Are Not Always Accurate

Other than doing lots of division, there are ways to see if a number is prime. However, this comes at a cost: Some composite numbers create false positives! This creates a type of integers called pseudoprimes, and different types of pseudoprimes emerge from different algorithms and parameters.

Think of the Fermat primality test, for example.

ap – 1 ≡ 1 (mod p)
is true if a and p is relatively prime and p is prime. However, some composite numbers also pass this test (like 341 and 561 for a = 2). Pseudoprimes are relatively rare among composite numbers, but it’s a thing to be aware of when conducting quick primality tests.

10. All Primes and Prime Powers are Solitary

What is a solitary number? It is an integer where the ratio between its sum of divisors and itself (abundancy index) is unique. Keep in mind that all numbers whose sum of divisors and itself are relatively prime are solitary.

If two values are relatively prime, the only way to produce this ratio with integers is by multiplying them. And here comes the problem: If n is a multiple of k and n does not equal k, the abundancy index of n is always greater than that of k. For instance, the abundacy index of 1002 is greater than that of 6 because part of 1002’s factors (167, 334, 501, 1002) already makes the abundancy index equal to that of 6, with the “extra” factors (1, 2, 3, 6) pushing the ratio further.

Prime powers are also solitary. Their sum of divisors is actually a repdigit in the associated base. For instance, the sum of divisors of 343 is 1 + 7 + 49 + 343 = 400, which is 11117. Since the extra one is added, these repdigits must be relatively prime to its base, which means that prime powers are solitary.

Conclusion

In this article, we’ve mentioned 10 interesting facts about prime numbers that you can easily prove or look up by yourself. Moreover, you can learn more about these fascinating integers in the references below. Besides, if we missed crucial points, feel free to mention them in the comments below.

References and Credits

  1. (n.d.). Fermat Pseudoprime. Retrieved August 9, 2021, from https://mathworld.wolfram.com/FermatPseudoprime.html
  2. Evelyn Lamb. (2019, April 2). Why Isn’t 1 a Prime Number? Retrieved August 9, 2021, from https://blogs.scientificamerican.com/roots-of-unity/why-isnt-1-a-prime-number/
  3. Great Internet Mersenne Prime Search. (n.d.). List of Known Mersenne Prime Numbers. Retrieved August 9, 2021, from https://www.mersenne.org/primes/
  4. The On-Line Encyclopedia of Integer Sequences. (n.d.). Indices of prime Fibonacci numbers. Retrieved August 9, 2021, from https://oeis.org/A001605

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