## Some Interesting Prime Numbers

Posted: April 14, 2018 in Mathematics

Prime numbers are the building blocks of all natural numbers. The Fundamental Theorem of Arithmetic states that every natural number greater than one is a prime itself or can be uniquely expressed as a product of primes. A prime number is a number that has one and itself as its factors, otherwise the number is composite. Although 1 has 1 and itself as its only factors, 1 is not considered as a prime. Due to the fact that defining 1 as a prime will destroy the uniqueness of the prime factorization of a number. Anyhow, there are a few prime numbers that I personally think they are interesting.

2 is the only even prime number

Any even number that is greater than 2 can be written as the product of 2 and a natural number greater than one so all even numbers greater than 2 are composite.

91 is NOT a prime number

I think almost all students in elementary school or even some in highschool thought 91 is a prime number. I am not kidding, if you are a teacher or a math tutor, you know what I mean. Almost 9 out of 10 students I encountered with always think 91 is a prime number. It could be that 91 isn’t in the times table? and it is odd and doesn’t end with a 5? and maybe it is obviously not divisible by 3 that makes people think 91 is highly unlikely a composite number? Or maybe people are too lazy to try to divide 91 by 7 and see for themselves.

1979 is the year I was born and it is a prime number

Yeah I am kind of proud to be born on the year that is a prime 🙂

2017 is the year which bitcoin increased 20 fold

It’s not because of bitcoin price has been increased from $1000 to$20,000 in the year 2017. The reason I think 2017 is an interesting prime number is that if you add up all the odd prime numbers from 3 all the way up to 2017, you will get a prime number as the result. 3 + 5 + 7 + 11 + 13 + … + 2017 = 283079 which is a prime number. You can check it with wolfram alpha. There are also some fun facts about 2017 I found on the internet.

• 20170123456789 is prime
• 2017π (round to nearest integer) = 6337 is prime
• 2017e (round to nearest integer) = 5483 is prime
• The sum of the cube of gap of primes up to 2017 is a prime number.
$(3-2)^3+(5-3)^3+(7-5)^3+\cdots+(2017-2011)^3$
• These three prime numbers are consecutive
$2011 = 2017+2-0-1-7$
$2017 = 2017 + 2 \times 0 \times 1 \times 7$
$2027 = 2017+2+0+1+7$

prime numbers that you can find in Pi

This is first 50 digits of Pi.

$\pi \approx 3.1415926535897932384626433832795028841971693993751 \ldots$

Couple years ago I was reading a book on recreational math and the author quoted that the first 12 digits of Pi (after rounding off) is a prime number. 314159265359 is a prime number. Of course you can always check it with wolfram alpha. So I found 3, 31, 314159, and 31415926535897932384626433832795028841 are also primes as well. There maybe more. I may wrote a script on maple to find more later when I have the time.

a prime that is unlucky with the devil

1000000000000066600000000000001 is a prime number. 666 is the number of a beast in the bible (I didn’t read the whole bible, that’s just what everyone is saying). The number 666 is considered as a devil which sits right inside in the middle of this prime number. And there are “13” zeros on the left and on the right of the number 666. Why 13 is unlucky? I don’t have a clue just go along with the majority.

12345678910987654321 is a prime number

This is no doubt one of the nice prime numbers even a 3 year old can remember.

a prime number that ends with “19” and “67”

1234567891011121314151617181967 is a prime number. This is a prime number that is easy to remember if you are a HKer and speak Cantonese because it is “19” and “67”. This number is very nice all you need to do is to write down the number 1, 2, 3, all the way up to 19 and then ends it with 67. 🙂

the largest known prime as of March 2018

I can’t end this article without writing down the largest known prime to human. As of March 2018 according to Wikipedia, the largest known prime is

$2^{77232917} - 1$

which is found by the GIMPS (Greatest Internet Mersenne Prime Search) in 2017. I am not gonna bother typing the number out because it is 23,249,425 digits long!