# The Great God Pan...

Mathurine
Posers and Puzzles 04 Jan '07 17:49
1. Mathurine
sorozatgyilkos
04 Jan '07 17:49
Try this one, then...

Find a positive integer (base 10) which has the following four properties:

(1) Each of the 10 digits (0 to 9) appears exactly once in the number.
(2) For each pair of digits whose sum is 9, the number of other digits positioned strictly between the pair is equal to the smaller digit of the pair.
(3) The sum of each pair of digits positioned at the same distance from opposite ends of the number is a prime number.
(4) The difference between any 2 adjacent digits is greater than 1.

The solution is unique.
2. BigDoggProblem
Just...the Dogg
04 Jan '07 22:161 edit
4 168 253 709

I didn't have much luck finding an elegant solution to this one. Instead, I built several "blocks" of varying length, based on property 2. Property 4 says that the 7 must be directly opposite either 0, 4 or 6. Using a spreadsheet, I shuffled the blocks and examined all combinations of 7 opposite 4, and none worked. I then tried 7 opposite 6, and got the answer.
3. 04 Jan '07 23:26
5 831 746 290?

BigDogg, 0 + 1 isn't prime.
4. BigDoggProblem
Just...the Dogg
05 Jan '07 00:221 edit
Originally posted by iMD
5 831 746 290?

BigDogg, 0 + 1 isn't prime.
1 used to be a prime number, under the definition "A number evenly divisible only by itself and 1".

Edit:

http://www.math.utah.edu/~pa/math/prime.html

"The reason why 1 is said not to be a prime number is merely convenience. For example, if 1 was prime then the prime factorization of 6 would not be unique since 2 times 3 = 1 times 2 times 3. A number that can be written as a product of prime numbers is composite. Thus there are three types of natural numbers: primes, composites, and 1." (emphasis mine)

I disagree with ruining a clear definition, like the one I gave, because of 'convenience'.
5. Mathurine
sorozatgyilkos
05 Jan '07 15:16
5831746290

is indeed the answer. Congrats., BDP !!
6. XanthosNZ
Cancerous Bus Crash
07 Jan '07 02:43
A prime number (or prime integer, often simply called a "prime" for short) is a positive integer p>1 that has no positive integer divisors other than 1 and p itself. (More concisely, a prime number p is a positive integer having exactly one positive divisor other than 1.) For example, the only divisors of 13 are 1 and 13, making 13 a prime number, while the number 24 has divisors 1, 2, 3, 4, 6, 8, 12, and 24 (corresponding to the factorization 24==2^3.3), making 24 not a prime number. Positive integers other than 1 which are not prime are called composite numbers.

Prime numbers are therefore numbers that cannot be factored or, more precisely, are numbers n whose divisors are trivial and given by exactly 1 and n.

The number 1 is a special case which is considered neither prime nor composite (Wells 1986, p. 31). Although the number 1 used to be considered a prime (Goldbach 1742; Lehmer 1909; Lehmer 1914; Hardy and Wright 1979, p. 11; Gardner 1984, pp. 86-87; Sloane and Plouffe 1995, p. 33; Hardy 1999, p. 46), it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own. A good reason not to call 1 a prime number is that if 1 were prime, then the statement of the fundamental theorem of arithmetic would have to be modified since "in exactly one way" would be false because any n==n.1. In other words, unique factorization into a product of primes would fail if the primes included 1. A slightly less illuminating but mathematically correct reason is noted by Tietze (1965, p. 2), who states "Why is the number 1 made an exception? This is a problem that schoolboys often argue about, but since it is a question of definition, it is not arguable." As more simply noted by Derbyshire (2004, p. 33), "2 pays its way [as a prime] on balance; 1 doesn't."

The definition of a prime number excludes 1 (by definition of course) and therefore the answer isn't the answer (by definition).
BigDogg may protest that 1 being excluded ruins the clear definition of a prime but including it ruins the clear definition of many other things (including as mentioned above the Fundamental Theory of Arithmetic).
7. BigDoggProblem
Just...the Dogg
07 Jan '07 18:04
Originally posted by XanthosNZ
A prime number (or prime integer, often simply called a "prime" for short) is a positive integer p>1 that has no positive integer divisors other than 1 and p itself. (More concisely, a prime number p is a positive integer having exactly one positive divisor other than 1.) For example, the only divisors of 13 are 1 and 13, making 13 a prime number, while th ...[text shortened]... many other things (including as mentioned above the Fundamental Theory of Arithmetic).
Here's the best part:

http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic

"In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers."

One minor problem: The number "1" disproves the theory! It is neither prime (having been robbed of its correct status), nor a product of primes.

So guess what? Yes, another clumsy exception to a so-called 'clear' definition. The wolfram site is careful to state that the fundamental theorem does not apply to the number 1. So much for convenience and clarity!
8. XanthosNZ
Cancerous Bus Crash
07 Jan '07 23:161 edit
Originally posted by BigDoggProblem
Here's the best part:

http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic

"In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers."

One minor problem: The number "1" disproves t ental theorem does not apply to the number 1. So much for convenience and clarity!
Find the number 1 on any list of prime numbers.

EDIT: We can go back and forth on this point (see http://www.geocities.com/primefan/Prime1ProCon.html for an example) but the point remains, almost without exception mathmaticians today do not consider 1 to be a prime.

And even if you cannot be convinced that it isn't you can clearly see that it would be very easy for someone else to think it isn't and therefore not find the solution to this problem.
9. 07 Jan '07 23:19
BDP doesn't handle being wrong well.
10. BigDoggProblem
Just...the Dogg
07 Jan '07 23:22
Originally posted by XanthosNZ
Find the number 1 on any list of prime numbers.
Oh, I know it's not there, for the same reason that we've got the QWERTY keyboard, the "English" measurement system (at least here in the US), the use of the letter "S" instead of "N" in chess problem magazines, etc. etc. We humans love to bow to convention and shirk logic.
11. BigDoggProblem
Just...the Dogg
07 Jan '07 23:23
Originally posted by dmnelson84
BDP doesn't handle being wrong well.
Hello again, fanboy.
12. XanthosNZ
Cancerous Bus Crash
07 Jan '07 23:27
Originally posted by BigDoggProblem
Oh, I know it's not there, for the same reason that we've got the QWERTY keyboard, the "English" measurement system (at least here in the US), the use of the letter "S" instead of "N" in chess problem magazines, etc. etc. We humans love to bow to convention and shirk logic.
I have no idea why the US has the Imperial measurement system, but the rest of the world seems to have logically decided that SI units are better. Maybe you should take that up with the US.

Calling 1 a non-prime is shirking logic? How's that? What logic is being followed to call 1 a prime?
13. BigDoggProblem
Just...the Dogg
07 Jan '07 23:31
Originally posted by XanthosNZ
I have no idea why the US has the Imperial measurement system, but the rest of the world seems to have logically decided that SI units are better. Maybe you should take that up with the US.

Calling 1 a non-prime is shirking logic? How's that? What logic is being followed to call 1 a prime?
Wait...why do you want to debate this now? You've been quoting source after source that claims it is a matter of definition. Why not just rest on that Ad Populum justification, if you're content with it?
14. XanthosNZ
Cancerous Bus Crash
07 Jan '07 23:33
Originally posted by BigDoggProblem
Wait...why do you want to debate this now? You've been quoting source after source that claims it is a matter of definition. Why not just rest on that Ad Populum justification, if you're content with it?
Matters of definition are always Ad Populum if you want to use that logic.

Why isn't 4 a prime number? I mean I view the definition of a prime as:
"All numbers 4 or less and any number greater than 4 which has only 1 and itself as factors."

And if you argue with me you are just appealing to the popular definition of a prime number!
15. 07 Jan '07 23:38
Originally posted by BigDoggProblem
Hello again, fanboy.
You wish.