From: jbaez@nevanlinna.mit.edu (John C. Baez)
Newsgroups: sci.math
Subject: The Little Book of Big Primes (repost)
Date: 27 Apr 92 12:39:41 GMT
I posted this before, but it seems that my posts haven't been making it
to the outside world (except those in sci.math.research?). If someone
outside MIT sees this, could they please send me email? If you don't
see it, silence will suffice. :-)
I recently checked out Paulo Ribenboim's "The Little Book of Big
Primes," new from Springer-Verlag, and decided to post some cute
tidbits. I am picking these not on the basis of theoretical interest,
just their appeal to the masses.
Recently, of course, there has been a lot of talk about primes on
sci.math. I recall some arguments over proving that there are
infinitely many. Some of you may want to look at Chap. 1 of Ribenboim's
book, titled How Many Prime Numbers Are There? He gives "five and two
half(!) proofs" that there are infinitely many.
If you know Fermat's little theorem, that x^(p-1) = 1 mod p (for x
nonzero mod p) when p is prime, it's obvious that for p prime
1^(p-1) + 2^(p-1) + ....... (p-1)^(p-1) = -1 mod p.
In 1950, Giuga asked if the converse was true: if
1^(n-1) + 2^(n-1) + ....... (n-1)^(n-1) = -1 mod n
is n prime? Since he was a wimp he only showed this held for
n < 10^{1000}. :-) In 1985 Bedocchi showed it was true for all
n < 10^{1700}. Ribenboim writes "A final atack on the problem is
needed. It may be not as simple as one would like. I did not try, but
I know of two good mathematicians who tried, but failed. Don't ask me
their names. It is all to their credit to hope to solve a problem like
this, which has, at first sight, no reason to have a positive or
negative answer."
Similarly, if n >= 5 is prime, the binomial coefficient
| 2n - 1 |
| n-1 |
equals 1 mod n^3. The converse, posed by Jones, is open.
Okay, now let me turn to some results which, as far as I can tell, are
the mathematical equivalent of junk food.
The largest prime all of whose digits are prime numbers is
7532(10^1104 - 1)(10^4 - 1) + 1,
discovered by Dubner in 1988. Now, just to prove to you how silly I am,
I'll print it out (with a little help from Mathematica):
753275327532753275327532753275327532753275327532753275327532753275327
532753275327532753275327532753275327532753275327532753275327532753275327
532753275327532753275327532753275327532753275327532753275327532753275327
532753275327532753275327532753275327532753275327532753275327532753275327
532753275327532753275327532753275327532753275327532753275327532753275327
532753275327532753275327532753275327532753275327532753275327532753275327
532753275327532753275327532753275327532753275327532753275327532753275327
532753275327532753275327532753275327532753275327532753275327532753275327
532753275327532753275327532753275327532753275327532753275327532753275327
532753275327532753275327532753275327532753275327532753275327532753275327
532753275327532753275327532753275327532753275327532753275327532753275327
532753275327532753275327532753275327532753275327532753275327532753275327
532753275327532753275327532753275327532753275327532753275327532753275327
532753275327532753275327532753275327532753275327532753275327532753275327
532753275327532753275327532753275327532753275327532753275327532753275327
532753275327532753275327533
The largest prime with one digit 1 and all other digits equal to 9 is
2 (10^3020-1) and was discovered by Williams in 1985. I think you can
sort of imagine what this one looks like so I won't print it out.
The largest prime all of whose digits are odd is 1358 (10^3821 - 1),
discovered by that same guy Dubner in 1988. I leave it to those
interested to calculate it.
The largest prime with all digits equal to 0 or to 1 is
10^641 (10^640 - 1)/9 + 1
It's sorta cute:
111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111100000
000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000001
Getting kind of worried near the end there, weren't you?
Now Ribenboim doesn't just say that these are the largest *known* with
these properties, but actually the *largest*. That, of course, would be
impressive, even though it's all mighty silly in my opinion. But
frankly, I have trouble believing that people know how to prove
something like that the above prime is the *largest* one with all 0's
and 1's in its decimal expansion.