— Non-Fiction —

Brilliant Numbers and a Few Sequences

ost numbers with names, such as squares, cubes, triangular numbers, abundant numbers, were given their monikers hundreds and perhaps even thousands of years ago. In this article, however, I would like to tell you about a new class of numbers (one perhaps even more important than those listed above) which received their name only a few years ago.

Brilliant numbers were coined by Peter Wallrodt and are defined as positive integers with exactly two prime factors of the same digital length (in decimal notation). For example,

123467 = 311 * 397

is a brilliant number since both of its prime factors have exactly three digits. Also

9785471483 = 98621 * 99223

is a brilliant since it has exactly two factors with five digits each.

The reason this class of numbers is important is that they often turn up in cryptographic settings and are used to test the current capabilities of prime factoring software, since the problem of factoring large numbers is considered computationally difficult. That is, no one knows a quick and easy way to break a number down into its prime factors (and large brilliant numbers are the most difficult of all since they have only two prime factors of equal decimal length).

Financial transactions that occur over the Internet are presently thought to be secure because they use public key cryptography algorithms (which I do not have enough space here to fully explain) that depend upon the difficulty of integer factorization to be safe. The first person who derives a method to instantly factor arbitrarily large integers will surely receive great amounts of fame in the mathematical world along with considerable monetary compensation as well.

Integer factorization was recognized as being important by the great mathematician Karl Friedrich Gauss (1777–1855) more than 200 years ago, and obviously it is still a difficult and significant problem. Here is what Gauss had to say about the importance of factoring numbers:

“The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length ... The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated."

—Karl Friedrich Gauss, Disquisitiones Arithmeticae.

According to Dario Alpern’s Web site (which can be found by Googling “brilliant numbers"), the largest brilliant currently known is 10¹⁴⁹ + 21553, which was found by Paul Leyland in July of 2006. It has these 75-digit factors:

196175124573517092034922493422470908957491913295260473203518969764669282857
and
509748624946200879944668835201017220741864841301635474800405920115982269129

Even though brilliant numbers have serious uses in the fields of cryptography and Internet security, they also possess recreational properties as well. For example, let’s consider the sequence of brilliant numbers that are also palindromes (a palindrome is a number that reads the same way forward and backward):

4, 6, 9, 121, 323, 737, 767, 949, 979, 989, 10201, 13231, 15151, 15251, 18281, 19291, 31313, 31613, 33233, 35653, 35953, 36863, 37073, 37673, 38383, 38683, 70307, 73537, 73937, 74447, 74947, 77677, 79297, 79897, 90109, 92129, 92329, 92629, 93839, ...

This is sequence A084350 in The On-Line Encyclopedia of Integer Sequences; and surely there are infinitely many numbers of this type. Can you find a 50-digit brilliant palindrome? How about a 100-digit brilliant palindrome?

Another exotic sequence that’s rather recreational in nature would be brilliant base-2 Fermat pseudoprimes. (A pseudoprime is a number that satisfies Fermat’s little theorem for a certain base, yet it’s still not prime.)

341, 1387, 2047, 2701, 31621, 42799, 49141, 49981, 60701, 83333, 88357, 90751, 104653, 129889, 130561, 150851, 162193, 164737, 219781, 226801, 241001, 249841, 282133, 1194649, 1678541, 2035153, 2134277, 2163001, ...

This is sequence A086837 in The On-Line Encyclopedia of Integer Sequences, and whoever proves that it is infinite will immediately become one of my mathematical heroes!

For a much more unusual sequence, consider finding the brilliants such that when they are concatenated with their 10’s complement (which must also be brilliant), the result is a prime number. (To get the 10’s complement of a number, simply put it into this function: f(n)=10^k – n, where k is the decimal length of the number (e.g. the 10’s complement of 1234 is 8766 because 10⁴ – 1234 = 8766 and 1234 + 8766 = 10000).) Could there be any numbers fitting the definition above? Yes. In fact, there are quite a few:

473, 779, 22331, 30353, 47573, 53237, 57599, 66767, 68021, 75953, 81797, 96023, 97133, 99221, 112661, 120983, 212519, 236849, 248687, 373097, 388511, 391649, 427319, 433793, 444359, 453689, 473699, 474689, ...

This is sequence A084629 in the O.E.I.S. and it is very strange!

Here is another exotic sequence that might be considered a reversal of the above definition: Primes such that when they are concatenated with their 10's complements (which also must be prime), the result is a brilliant number:

11, 257, 509, 929, 2243, 2897, 3911, 4409, 7121, 9413, 10739, 11411, 13217, 17783, 19319, 20849, 21377, 32507, 32957, 41729, 47279, 48761, 87041, 93083, 93263, 93911, 95027, 95603, 96221, ...

Highly unusual sequences indeed.

So now that you know what brilliant numbers are and understand a few of their properties, you may want to give a short lecture on them at your next barbecue or family reunion to demonstrate your numerical nerdiness. If you do, you may just inspire your friends and family to start searching for weird numbers themselves!