Well, here we are again asking ourselves life's never ending questions: "Should I tell the world?" "Did I give up?" "Should I go on and on about what I did to try and solve it?" "Was I able to solve it?" "How much time and how many sessions did I give to it?" "If I didn't solve it on my own, what did I do to finally solve it?"
A recent comment on yofx makes me want to contribute but also makes me want to get you thinking some more. So ... here goes.
I didn't solve it but I now know the answer. Well, I now know an answer. I am pretty sure it's the answer, but ya never know.
I tried Roman Numerals. I tried adding two multi-digit numbers. I tried subtracting two multi-digit numbers. I tried using the multiply symbol (i.e. x) and a bunch of factors that only used two different digits. I even tried getting cute and saying things like "3 symbols is a bit excessive because 1101110101111100000000 is 3628800" in base 2. I then almost got real cute and was going to use the full three symbols and convert 3628800 to base 3, but I got lazy when I found my calculator didn't do base 3 so I dropped that idea. Of course I factored the number and got hopeful when I started seeing things like
but that damn 5 and a few other insurmountable things made me realize I was barking up the wrong tree.
Of course there was always Google, the last resort. I even did stuff like "seconds in six weeks using 3 symbols."
Then I got lucky!
Of course I wish I could have nailed this on my own. I was inching up on it. The answer made me think of the novel variation on the classic proof by Euclid that there are infinitely many primes (as modified by Hofstadter in his Godel, Escher, Bach). Jeff, whose last name I don't know, a peer tutor, told me about this the other day.
That last paragraph is probably more misleading than helpful. But, if you really have a lot of time on your hands and this problem is driving you bananas, you could always go to the Tunxis library, find the book, check the index, read the page and, not only will you get new insight into this beautiful, as Erdos said, "straight from The Book" proof, but you might get an idea on how this problem gets solved.
Hoping I have not confused you more than I have helped you, Good Luck!
I've never read Godel, Escher, Bach, but I love that proof. I always think that one of the coolest things about math is that you can prove something is impossible or will never happen. This is something that completely sets apart the epistemology of math from that of science.
Here, essentially, is Hofstadter's alternative to Euclid's proof. (I'm not sure Hofstadter came up with this himself, but he does have it in his book.)
Suppose there were a largest prime number. Call it N. Now consider N! + 1. Clearly, N! + 1 does not have any number between 1 and N as a divisor. This means that either a) N! + 1 is prime, or b) N! + 1 has a prime divisor greater than N. In either case, we obtain a contradiction. Thus, there is no largest prime number.
BTW, in case you missed it, the Fundamental Theorem of Arithmetic is behind possibility b) above.
Now, we really need to focus and remember that we're still trying to solve Steve's original problem. The above proof uses factorials.
5 comments:
Well, here we are again asking ourselves life's never ending questions: "Should I tell the world?" "Did I give up?" "Should I go on and on about what I did to try and solve it?" "Was I able to solve it?" "How much time and how many sessions did I give to it?" "If I didn't solve it on my own, what did I do to finally solve it?"
A recent comment on yofx makes me want to contribute but also makes me want to get you thinking some more. So ... here goes.
I didn't solve it but I now know the answer. Well, I now know an answer. I am pretty sure it's the answer, but ya never know.
I tried Roman Numerals. I tried adding two multi-digit numbers. I tried subtracting two multi-digit numbers. I tried using the multiply symbol (i.e. x) and a bunch of factors that only used two different digits. I even tried getting cute and saying things like "3 symbols is a bit excessive because 1101110101111100000000 is 3628800" in base 2. I then almost got real cute and was going to use the full three symbols and convert 3628800 to base 3, but I got lazy when I found my calculator didn't do base 3 so I dropped that idea. Of course I factored the number and got hopeful when I started seeing things like
this
but this uses one more symbol than the allowed 3.
I really got hopeful when I found
42*24*4*4*225
but that damn 5 and a few other insurmountable things made me realize I was barking up the wrong tree.
Of course there was always Google, the last resort. I even did stuff like "seconds in six weeks using 3 symbols."
Then I got lucky!
Of course I wish I could have nailed this on my own. I was inching up on it. The answer made me think of the novel variation on the classic proof by Euclid that there are infinitely many primes (as modified by Hofstadter in his Godel, Escher, Bach). Jeff, whose last name I don't know, a peer tutor, told me about this the other day.
That last paragraph is probably more misleading than helpful. But, if you really have a lot of time on your hands and this problem is driving you bananas, you could always go to the Tunxis library, find the book, check the index, read the page and, not only will you get new insight into this beautiful, as Erdos said, "straight from The Book" proof, but you might get an idea on how this problem gets solved.
Hoping I have not confused you more than I have helped you, Good Luck!
- Lee
Obfuscated hint:
Tack on another thirty-six million two hundred eighty-eight thousand seconds and you only need two symbols for the sum.
I've never read Godel, Escher, Bach, but I love that proof. I always think that one of the coolest things about math is that you can prove something is impossible or will never happen. This is something that completely sets apart the epistemology of math from that of science.
Here, essentially, is Hofstadter's alternative to Euclid's proof. (I'm not sure Hofstadter came up with this himself, but he does have it in his book.)
Suppose there were a largest prime number. Call it N. Now consider N! + 1. Clearly, N! + 1 does not have any number between 1 and N as a divisor. This means that either a) N! + 1 is prime, or b) N! + 1 has a prime divisor greater than N. In either case, we obtain a contradiction. Thus, there is no largest prime number.
BTW, in case you missed it, the Fundamental Theorem of Arithmetic is behind possibility b) above.
Now, we really need to focus and remember that we're still trying to solve Steve's original problem. The above proof uses factorials.
Woops!
The version I know only uses the product of the primes from 2 to N, but to the same effect.
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