Date: Dec 7, 2012 12:47 AM
Author: James Waldby
Subject: Re: No Putnam spoilers please
On Thu, 06 Dec 2012 00:10:22 +0200, Phil Carmody wrote:

> Adding rec.puzzles

>

> William Elliot <marsh@panix.com> writes:

>> On Sat, 2 Dec 2012, Bart Goddard wrote:

>> > William Elliot <marsh@panix.com> wrote in

>> >

>> > > How do we know if a question is a Putnam question?

>> >

>> > It'll have the number 2012 in it somewhere.

>>

>> "What's 5^2012 mod 7?" is a Putnam question? ;-}

>

> I come with Putnam-related news from the future. My future self

> remains interested in Putnam questions, and so has seen them all

> for many years to come. Alas, in bringing the question back in

> time, some of the information has become lost.

>

> He/I says that one of the questions is:

> "What is b^y mod y?"

> where 'b' is a small number, and 'y' is the year of that Putnam

> competition. He also provided me with the answer, which was another

> year from the future where the Putnam took place.

Presumably the answer x is such that b^y mod y == x and b^z mod z == x

where z = y+1, and x > 2012, ie, x is in our present future, not the

future relative to y or z, because x is less than both of them.

> Strangely, he sent me this message twice, a year apart.

>

> What puzzles me is when were these messages were sent, and what the

> actual questions were before the information loss...

There seem to be at least a hundred such year-pairs in the next

thousand years; for example,

30^2469 mod 2469 == 2310 and 30^2470 mod 2470 == 2310

32^2312 mod 2312 == 2024 and 32^2313 mod 2313 == 2024

51^2974 mod 2974 == 2601 and 51^2975 mod 2975 == 2601

832^2041 mod 2041 == 2028 and 832^2042 mod 2042 == 2028

although perhaps the value of b in the last case isn't "small".

More distantly,

2^9053 mod 9053 == 8632 and 2^9054 mod 9054 == 8632

3^66704 mod 66704 == 20673 and 3^66705 mod 66705 == 20673

4^7781 mod 7781 == 4096 and 4^7782 mod 7782 == 4096 = 2^12

5^7015 mod 7015 == 4745 and 5^7016 mod 7016 == 4745

6^24655 mod 24655 == 7776 and 6^24656 mod 24656 == 7776 = 6^5

7^7203 mod 7203 == 2401 and 7^7204 mod 7204 == 2401 = 7^4

8^58482 mod 58482 == 29242 and 8^58483 mod 58483 == 29242

9^8207 mod 8207 == 6993 and 9^8208 mod 8208 == 6993

10^9233 mod 9233 == 4618 and 10^9234 mod 9234 == 4618

11^17347 mod 17347 == 14641 and 11^17348 mod 17348 == 14641 = 11^4

9^66428 mod 66428 == 6561 = 3^8

9^66429 mod 66429 == 6561

9^66430 mod 66430 == 6561

--

jiw