What Are Google Primes?

A moderately famous number (as far as numbers are concerned anyway) is 71077345. Performing a Google search on this number brings up many explanations that run similar to this: Enter 71077345 into a calculator, flip the calculator over, and you will see the word "ShELLOIL" spelled.

After discovering this amusing factoid I thought it would be fun to experiment with the idea of finding other numbers that spelled out certain words when they are read upside down on a calculator; and to see if they had any "interesting" mathematical properties as well.

A few hours of exploration soon revealed that the number 379009 spells "GOOGLE"[1] when turned upside down on a calculator. This number also happens to be prime (an integer with no divisors except 1 and itself). I sent this curiosity to the good people at Google Labs, along with an explanation and a suggestion that they put the number up on their main search page sometime, since they occasionally do such things around major holidays, but I never received a reply.

Then I became curious about 379009 again. I wondered if I would find more primes by allowing as many zeros as possible between the two 9s. That is, I defined a simple function, gp(n) = 379*10^n +9, and used the free primality testing program PFGW[2] to search for values of n that would make gp(n) a prime number.

It turns out there are many Google primes. When n = 3, 6, 8, 9, 37, 44, 67, 111, 157, 289, 1256, 1602, 2410, 2482, 2868, 3824, 3891, 6595, 8984, 9318, and 10274, gp(n) is a probable prime. I searched up to n = 15,000 with no more values found. Because these numbers are not of an easily provable form, I had to use the online ECM factorization applet written by Dario Alpern[3] to prove that all values up to n = 1256 are actually prime numbers. Note that gp(1256) is a "titanic prime"[4] since it has over 1,000 digits.

Now that Google Primes have been defined, I think the question of whether there are infinitely many will remain unanswered for quite awhile. When do you think number theory will be sophisticated enough to handle questions such as the former? Can you find a larger Google probable prime, or prove one of the larger probable prime values listed above?

Imagine a calculator with an infinitely long display, enter the number 379000...(insert as many zeros here as you like)...9 and it will always spell Go...ogle when flipped upside down on a calculator.

But what about textual primes? A prime with a word or phrase "pictured" in its decimal expansion, after certain digits have been bolded. That's easy. See the picture that goes along with this article.

References

1. Google Search Engine, http://www.google.com/
2. PrimeFormGW (PFGW), Primality-Testing Program Discussion Group,
http://groups.yahoo.com/group/primeform/
3. Dario Alpern, Factorization Using the Elliptic Curve Method Applet,
http://www.alpertron.com.ar/ECM.HTM
4. Chris Caldwell, The Prime Glossary, Titanic Prime,
http://primes.utm.edu/glossary/page.php?sort=TitanicPrime