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The IRS's Favorite Mathematical Law

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When it comes to catching tax cheats, the IRS has more than just federal law on its side. The agency’s arsenal also includes a mathematical truth known as Benford’s law. Armed with this law, the IRS can sniff out falsified returns just by looking at the first digit of numbers on taxpayers’ forms.

While most Americans wouldn’t put it past the IRS to use black magic, the truth behind Benford’s law is far from mystical. In 1938, GE physicist Frank Benford undertook a comprehensive study of numbers and how they occur. His findings mirrored the discovery of American astronomer Simon Newcomb, who had undertaken similar research in 1881. Benford found that when it came to naturally or socially generated data, the distribution of the first digit in a series of numbers is not uniform.

In analyzing 20,000 sets of numbers from a variety of sources—numbers from a newspaper, population figures, American League baseball stats—Benford found that a whopping 30 percent of the numbers in his sample had one as their first digit. The numeral two turned up in the first position 18 percent of the time, and three occurred 12 percent.

There’s a simple explanation for what Benford observed. In the number set 0 to 99, 11 percent of the numbers start with 1, and 11 percent start with each digit from 2 to 9. In the number set 0 to 199, over half of the numbers start with 1, and less than 6 percent start with 2 to 9. In the number set 0 to 299, 37 percent start with 1 and 37 percent start with 2, and the numbers 3 through 9 start 3.7 percent each. This situation goes on forever, so over a large enough data set, the distribution of leading digits follows a predictable pattern. The bigger the integer, the less likely it is to be the first digit in a data set.

How does the IRS use this distribution? Many folks who happen to fudge a bit on their tax returns or expense reports call attention to their creativity by using too many dollar amounts that start with an eight or nine (the least common integers found in the first position) and not enough that start with numeral one. Savvy CPAs know what to look for, and many computer systems that tabulate figures are also programmed to catch any suspicious strings of numbers.

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Art
A Man-Made Mountain in Finland Serves as an 11,000-Tree Time Capsule
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In 1982, the conceptual artist Agnes Denes set out to make a mountain. After a decade of work, she made it happen. In 1992, the Finnish government announced that it would sponsor Denes’s Tree Mountain—a 125-foot-tall manmade mountain built on top of a former gravel pit, designed to serve as part time capsule, part ecological recovery project.

Tree Mountain — A Living Time Capsule was constructed on the site of a former gravel pit near Ylöjärvi, Finland between 1992 and 1996. The artificially constructed landmass stands 125 feet tall, almost 1400 feet long, and more than 885 feet wide. (The top image of the triptych above shows the mountain in 1992 and the bottom image in 2013.) The forest planted on it forms a precise mathematical pattern Agnes designed based on the golden ratio-derived spirals of sunflowers and pineapples. From above, the oval mountain looks like a giant fingerprint made up of whorls of trees.

The project was never intended to just be aesthetically pleasing. Envisioned as a way to rehab land destroyed by mining, the trees are meant to develop undisturbed for 400 years, creating what will eventually be an Old Growth forest that can reduce erosion, provide wildlife habitats, and boost oxygen production.

And it was a communal effort. The roughly 11,000 pine trees were planted by different individuals who then became the custodians of those trees. Each received a certificate declaring their ownership for the project’s full term of 400 years. They can pass along this ownership to their descendants or to others for as many as 20 generations. These custodians (which include former UK prime minister John Major and former Icelandic president Vigdís Finnbogadóttir) are even allowed to be buried under their trees. But the trees can never be moved, and the mountain itself can’t be owned or sold off for those 400 years.

A triptych of images of Tree Mountain
Tree Mountain - A Living Time Capsule - 11,000 Trees, 11,000 People, 400 Years (Triptych) 1992-1996, 1992/2013
Copyright Agnes Denes, courtesy Leslie Tonkonow Artworks + Projects, New York

“Tree Mountain is the largest monument on earth that is international in scope, unparalleled in duration, and not dedicated to the human ego, but to benefit future generations with a meaningful legacy,” Denes writes. It “affirms humanity's commitment to the future well being of ecological, social and cultural life on the planet.”

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This Just In
New Largest Known Prime Number Has More Than 23 Million Digits
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Prime numbers come in all sizes: They go down to single digits and grow infinitely larger. But calculating the exact quantity of the largest prime numbers in existence takes serious time and effort. Now, thanks to help from a volunteer and his computer, the Great Internet Mersenne Prime Search (GIMPS) has identified the newest largest prime number we know of.

The prime number has 23,249,425 digits, surpassing the previous record holder by 1 million digits. It can be written as 277,232,917-1 or M77232917. Like other prime numbers, the quantity can only be divided by one and itself. But unlike some smaller primes, this one joins a special category called Mersenne primes.

Mersenne primes are found by calculating numbers to the second power and subtracting the value of one from the total. Only 50 prime numbers have been found this way, and a lot of computing power is required to uncover them.

Since 1996, GIMPS has been crowdsourcing computers to discover larger and larger prime numbers. Anyone can download their program and dedicate their unused processing power to churning out algorithms in search of the next record breaker. Volunteers whose computers successfully identify a new prime number are eligible for a cash reward of up to $3000.

The most recent winner was Jonathon Pace, a 51-year-old electrical engineer from Tennessee. His computer calculated the number M77232917 on December 26, and its prime status was independently verified by four separate computers.

GIMPS is constantly outdoing itself, with the previous largest prime announced just two years ago. If you'd like to join the effort, their prime-hunting software is free to download. But don't expect instant results: Pace was volunteering with GIMPS for 14 years before his altruism paid off. 

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