PayPal's recent accidental crediting to the account of one Chris Reynolds of $92,233,720,368,547,800.00 (that's more than 92.2 quadrillion dollars, in scientific notation $9.22 x 1016) attracted worldwide press attention. The Huffington Post was probably not alone in using "astronomical" to describe the sum. And indeed, if you could travel one mile with each of those dollars, your trip would reach 15,689.7 lightyears, a little more than halfway to the center of our galaxy. And if each of those dollars were a year, Reynolds' number (not to be confused with Reynolds number) would be 6.7 million times the present age of our universe (13.77 billion years). If you spent one Reynolds dollar every 5 seconds since the Big Bang you'd still have around a half quadrillion left, almost enough for a down payment on a studio apartment here in New York City.
Even the tired old analogies give impressive results. A stack of 92.2 quadrillion dollar bills, each dollar being 0.0043 inches thick, would be 6.26 billion miles high. That's 67.3 times the distance between the Sun and Earth (that is, 67.3 astronomical units or AU). That stack of dollars would reach way past Pluto into the orbit of the dwarf planet Eris. A beam of light would take 9.33 hours to travel from one end of the stack to the other. Amazingly, the lowly—and at $865 million ($8.65 x 108), far less expensive—Voyager-1 spacecraft, launched in 1977, has gone nearly twice this distance (around 126 AU) and has just left the Solar System (or not, depending on how you interpret the data we're getting). If instead of making a stack of bills you placed each dollar end to end, light would take 555.3 days to travel from the first to the last bill since they would extend to 1.5 light years. Impressive as that sounds, it isn't even halfway to the next nearest star (Proxima Centuri at 4.24 light years). I'll have more on Voyager-1 and its attached LP record in a future post.
Considered as actual currency, $9.22 x 1016 is way more dollars than have ever been, anywhere in the Solar System. And since it's quite unlikely that any aliens are using units called "dollars," by extension PayPal's accident is more dollars than have ever been on any class-M planet in the Alpha Quadrant (to lapse into Star Trek terminology) or indeed on any planet in the entire Milky Way. And this very likely holds true all the way up the scale of cosmic structure, to the Local Group (of galaxies) up through the Virgo Supercluster and all the way up to the entire observable universe. PayPal's mistake is 18.4 times the $5 quadrillion total value of Earth, so $92.2 quadrillion is pretty much an impossible figure to obtain by any conceivable financial transaction on Terra Firma. This alone should have set off alarm bells at PayPal.
Maybe an examination of Reynolds' number itself might provide some clues as to what went wrong. I'll ignore the fact that his PayPal account statement was in both dollars and cents, and I'll just discuss the whole-dollar amount, which is a 17-digit base-10 integer. There is no obvious pattern in the digits, as if somebody were typing in a phone number to spell out a message. And I also find it unlikely that those digits were obtained by somebody accidentally leaning on a keyboard. The chance of a computer coming up with this number at random is at best 1 in 92.2 quadrillion; you are around 5 billion times more likely to win the PowerBall lottery. Then again in both cases the odds might be considerably improved if the random number algorithm has been subverted by the NSA (more on them in another upcoming post). [Yo! NSA compuers scanning this! Sup in Fort Meade?]
Examining the number as a number and not merely a set of characters, it ends in zero and so is even and not a prime number. But discovering its prime factors using web-based tools is tricky since some of the on-line prime-factor calculators that come at the top of a Google search for can't handle 17-digit numbers. And at least one of the calculators seemingly accepts 17 digits but then gives an incorrect answer. Factoring large numbers is inherently difficult, a fact that is taken advantage of by cryptographic systems in widespread use. Appropriately enough, the first correct factorization I got (2 × 2 × 2 × 3 × 5 × 5 × 7 × 21960409611559, a disappointingly dull result) was obtained from a Java applet that uses some of the same "elliptic-curve" techniques employed by modern cryptosystems, including some using algorithms approved by the NSA.
Humans can get all excited about prime numbers, but typical computer hardware could hardly care less (though it's a safe bet that the NSA's codebreaking processors do care, a lot). All that the circuits know about are voltage levels signifying binary numbers, prime or otherwise. Pehaps conversion of Renyolds' number into binary digits might prove enlightening: 101000111101011100001010001111010111000010100011111011000. Or maybe not. Lets try the even more geeky hexadecimal (base-16) notation often used by programmers, where 0 to 9 decimal equals 0 to 9 hex, and 10 to 15 decimal is A to F in hex. Decimal 92233720368547800 = hexadecimal 147AE147AE147D8.
Wow, a pattern! (In retrospect, you can detect it in the binary.) Let me quickly point out that this pattern is not nearly on the transcendent level as the numerical revelation at the end of Carl Sagan's Contact (the novel, not the movie). At least I don't think it is. It could be nothing but a coincidence (remember I ignored such possibly relevant factors as the digital representation of cents). Nonetheless, duplicating bit patterns, or moving them from side to side within a binary number, is something digital computers can do easily, since such functions are literally built into the fundamental instructions of a computer (the processor's microcode). So a Reynolds-type error could come from a programming error or even a circuit error caused by the processor or memory chips being hit by subatomic particles from a nuclear decay or from cosmic radiation. Or perhaps the number was mistakenly shunted to Reynolds' account from a buggy version of the recently-launched Pay-Pal Galactic system, which will enable off-planet transactions (including paying for travel on PayPal's retro-looking space ship pictured above) and which probably should allow transactions into the quadrillions. This would be especially important when buying and selling whole planets—and when dealing with the Ferengi.