The tribute...
This puzzle is a tribute to one of the great geocachers in South-East Queensland, for whom I have tremendous admiration and respect. He can often be seen zooming around in Suzee`, climbing a mountain, or cracking open an ammo can. This one is for you, Pprime (P`).
* Update 6/08/2020 - Congratulations Pprime (P`) on a very fitting FTF. Couldn't have gone to a better person.
Prime numbers...
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
The property of being prime is called primality. A simple but slow method of checking the primality of a given number \(n\), called trial division, tests whether \(n\) is a multiple of any integer between 2 and \(\sqrt{n}\). Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018 the largest known prime number has 24,862,048 decimal digits.
There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm.
The puzzle...
You are looking for 15 digits, from 0 to 9. Below is a PRIME number of statements about the digits you seek:
- The sum of the last three east digits minus the sum of the first three east digits is a PRIME number.
- There are a PRIME number of east digits which are themselves PRIME.
- The sum of the first three south digits minus the sum of the last three south digits is a PRIME number.
- There are a PRIME number of south digits which are themselves PRIME.
- The sum of the south digits minus the sum of the east digits is a PRIME number.
- 0 and 1 are, by definition, not PRIME. However, a PRIME number of the 15 digits are either 0 or 1.
- Either the first and last south digits are both PRIME and both equal, or, the first and last east digits are both PRIME and both equal.
It is a difficult puzzle, but if you've done the other two Numbers puzzles, you will at least have a head start. Remember to think rationally and have a little faith in yourself.
The cache...
Please check the attributes! This cache is a giant black bison tube up a tree. A long way up a tree. You won't need ropes, but you will need to be a competent tree climber, and assess the risks of climbing for yourself. Look for a tree with a bit of a lean.
And finally...
Good luck, and happy caching!