Thursday, July 25, 2013

Miss Liberty 4: Mystifying Dr. Memory

            Miss Liberty and I were standing outside a gypsy doctor’s tent. The sign at the door read:

Enter here.
!!! SEE !!!
the amazing DOCTOR MEMORY!!
Sees  all!
Hears all!
Tells all!
- prognostications and divinations a specialty -
The Key To The Future Is Yours!

            We entered the tent. Bells jingled.
            Inside was Dr. Memory, sitting at a small circular desk. Dr. Memory was wearing a star-spangled robe and a turban. On the desk were two glowing crystal balls.
            “Ahh!” Dr. Memory cried as we entered. “The balls are clearing again! The left ball is the Sun, the right ball is the Moon. Put what you want between them, and your future begins.”
            Liberty put a $0 bill in between.
            “Thank you,” Dr. Memory said greasily. Then he closed his eyes  and said. “Alas, I see the seal of doom upon you! You shall soon meet your unmaker!”
            Liberty asked, “When?”
            Dr. Memory said, “After I answer your questions. First question, please.”
            “I assume that you have heard of the Pythagorean Theorem,” said Liberty.
            Dr. Memory touched his turban. One of the crystal balls clouded over; then it showed the following formula:
                        a2 + b2 = c2.
            She continued, “And I assume you know the solutions to that equation, in the integers.”
            The other crystal ball clouded over, then showed:
                        a          =          (m2 - n2) k,
                        b          =            (2mn)   k,      
                        c          =          (m2 + n2) k,
            “And those are all the integer solutions,” said Dr. Memory.
            “Very exact!” said Miss Liberty. “But I was wondering about that equation there.” She pointed to the first crystal ball. “Suppose we change the exponent to a general integer n.” The first crystal ball clouded over, then showed:
                        an + bn = cn
            “I was wondering about its solutions for n bigger than 2. Are there any?”
            “That equation has no solutions in integers when n is greater than or equal to three.”
            Miss Liberty asked, “You’re sure of this.”
            “I have memorized the proof,” said Dr. Memory. “Next question, please.”
            Miss Liberty said, “Well, I noticed a funny pattern. Take a look at this.” She pointed at the second crystal ball. It clouded over, then showed:
                          4 = 2+2
                          6 = 3+3
                          8 = 3+5
                        10 = 5+5
                        12 = 5+7
                        14 = 7+7
                        16 = 5+11
            Miss Liberty said, “Note how all the numbers on the left are even.”
            Dr. Memory said, “I so note.”
            She continued, “And note how all the numbers on the right are prime.”
            Dr. Memory said, “I so note.”
            Miss Liberty said, “So I wonder if this pattern continues. Is every even number bigger than 2 a sum of two primes?”
            “Every even integer from four to a thousand is a sum of two primes,” said Dr. Memory.
            “So far, so good.”
            “I have proven that every even integer from four to a million is a sum of two primes.”
            “This looks like a trend!” said Liberty.
            “Now I have proven it for every even integer from four to a billion.”
            “Good work!”
            After thinking awhile, Dr. Memory said, “Now I have a proof for four to a trillion.”
            “You’re getting closer! Now on to a quadrillion! A googol! A googolplex!”
            Dr. Memory said, “I must meditate further on this question.” He closed his eyes, then re-opened them. “Next question, please.”
            Miss Liberty pointed at the second crystal ball; it clouded over and showed:
                        zeta(s) = 1/1s + 1/2s + 1/3s + 1/4s + ...
            “Zeta is defined on the complex numbers, you understand,” said Miss Liberty.
            “I understand,” said Dr. Memory.
            “I was wondering about the real part of zeta’s roots.”
            After thinking awhile Dr. Memory said, “I have found a root of the zeta function. Its real part is one-half. I have found another root. Its real part is also one-half. I have found all roots within one hundred units of the real axis. All their real parts are one-half.”
            “Another trend! So my third question is: do all of zeta’s roots have real part one-half?”
            “Every root within one thousand units of the real axis has real part equal to one-half.”
            “Almost a proof!”
            “Within one million units.”
            “Closer still!”
            “I am not done yet,” said Dr. Memory. “I must meditate further on this question.” And he closed his eyes again.
            Miss Liberty sang;       “Oh where are the zeros of zeta of s?
                                                I cannot inform you, I truly confess!
                                                Their real part is always one-half, I suppose;
                                                But no-one can prove it, so nobody knows!”

            Then the lights went out.
            In the sudden darkness we heard Dr. Memory say, “This program is jammed. Consult supervisor.”

            * click *

            Thus Miss Liberty mystified Doctor Memory with Goldbach’s Conjecture and the Riemann Hypothesis.
            Doctor Memory said, “You now have Full Access to #2. Beware, Master! She’s a glitch! Burn her!”       

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