Calculator Fun and Games: Premium Prime Numbers

 PRIME NUMBERS

 Calculator Fun and Games is a maths puzzle book that’s worth its salt: it has a section on prime numbers, as any good maths book does.

Prime numbers are only divisible by themselves and 1, they have no other factors.  1 is not a prime number (by definition), but 2 is, making 2 the only even prime number.  The study of prime numbers dates back thousands of years. Some of the earliest records come from the ancient Greeks, particularly the mathematicians of Pythagoras' school (around 500 BC to 300 BC), who explored numbers for their mystical properties.  I recently looked at Pythagoras's Triples - there's always an overlap between different parts of maths.

One of the most significant contributions came from Euclid (circa 300 BC), who proved that there are infinitely many prime numbers. His work in Elements laid the foundation for number theory, including the Fundamental Theorem of Arithmetic, which states that every integer can be uniquely factored into prime numbers.  That's worth its own investigation, and is a puzzle in itself.

Calculator Fun and Games lists the prime numbers up to 1000, just for good measure.  But is there any relationship between them?  Are they connected?  Are there ways of finding (or generating) them?  And can all this be done with a humble calculator?  Maybe one day.

However, the book points out an interesting fact:  if you

Take a prime number (greater than 3)
Square it
Add 14
Divide by 12

Then the remainder is always 3.

For example:
5² = 25
25 + 14 = 39
39/12 = 3, with a remainder of 3

Let’s take a larger example: 577
577² = 332929
332929 + 14 = 332943
332943 / 12 = 27745 remainder 3

Alternatively, x² + 11 always divides exactly by 12.  I’m not sure why Ben Hamilton decided on adding 14… maybe he likes the remainder 3?

Does this apply in reverse?  If I take a number, and subtract 11 and take the square root, do we always get a prime? 

No.  Even if the square root is an integer, it doesn’t mean that the starting number was a prime.  For example, 92-11 = 81, and the square root of 81 is 9.  Only a subset of a = (SQRT(b-11)) will give a as a prime.

TWIN PRIMES

Twin primes are primes with a difference of two; for example, 11 and 13, or 17 and 19.  These are rarer than prime numbers, but it still seems that there will be an infinite number of twin primes.  An example of a larger twin prime is 971 and 973; while the largest known twin primes have 388,342 digits:  they are:  2996863034895×2^1290000±1.

Not a number you’d fit on your calculator.  The largest prime number which will fit on an eight-digit calculator (with a good old-fashioned LCD display) is 99,999,989.

Looking at all those nines, I’d like to play Over and Out with it, and see how long it would take to get it down to zero! 😊

Other recent Calculator Fun and Games articles:

Snakes and Ladders (Collatz Conjecture)
Crafty Calculator Calculations (numerical anagrams, five digits)
More Multiplications (numerical anagrams, four digits)
Over and Out (reduce large numbers to zero in as few steps as possible)