Perfect, Friendly Numbers: Mathematical Puzzles That Still Challenge
Published by The mathematicians of ancient Greece attributed characters to numbers and awarded some of the status of perfection. For Euclid, one of the finding fathers of modern mathematics, a perfect number was one that equaled the sum of its own divisors – that is numbers that will divide into it without leaving a remainder. The first perfect number is 6: its divisors are 1, 2, and 3, 14 which add up to 6. The second is 28 (1 + two + four + seven + 14). The Greeks knew only two other perfect: 496 and 8,128.
Perfect, Friendly Numbers – Mathematical Puzzles That Still Challenge
What puzzle has been worked on continually for more than 2,000 years, has no practical application, and will probably never be solved? The answer: the search for the definitive list of perfect and friendly numbers.
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The mathematicians of ancient Greece attributed characters to numbers and awarded some of the status of perfection. For Euclid, one of the finding fathers of modern mathematics, a perfect number was one that equaled the sum of its own divisors – that is numbers that will divide into it without leaving a remainder. The first perfect number is 6: its divisors are 1, 2, and 3, 14 which add up to 6. The second is 28 (1 + 2 + 4 + 7 + 14). The Greeks knew only two other perfect: 496 and 8,128.
Quantum Jump
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More than 1,500 years later, in the 15th Century, the existence of a fit perfect number was announced 33, 550,336. Four additional perfect numbers were discovered in the next three centuries. But such was their rarity that in 1811 the mathematician Peter Barlow confidently stated the ninth perfect number, one with 37 digits, “is the greatest that will ever be discovered…It is not likely that any person will attempt to find one beyond it.” But in 1876 Barlow was proved wrong when a 10th perfect number was found – one with 77 digits.
Today – the list has been greatly extended and is still constantly growing. The largest known perfect number, the 27th, has a staggering 26,790 digits; it was revealed, with the help of a computer, in 1979. But many puzzling aspects of perfect numbers remain. Why, for example, are all the known perfects even numbers? Is there a largest perfect number? Or will there always be others to find?
Friendly Numbers
Related to perfect numbers are amicable or friendly numbers. The Greek mathematician Pythagoras regarded two numbers as friendly if each was the sum of the other’s divisors. The Greeks were aware of just one such pair, 230 and 284. The divisors of 220 (1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110) add up to 284, while the divisors of 284 (1, 2, 4, 71, 142) add to 220.
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Not until 1636 was another pair of friendly numbers – 17,296 and 18,416 – discovered by the French mathematician Pierre de Fermat. However, by the middle of the 19th century, the number of known friendly pairs totaled more than 60. Incredibly, the second-lowest pair of all had been missed. In 1867 a 16-year-old Italian, Nicolo Paganini, demonstrated that 1,184 and 1,210 are friendly.
There are questions associated with friendly numbers too. All known examples consist of either two odd or two even number possible? Why all the odd friendly are numbers multiples of there?
Mathematicians will continue to puzzle over the mysterious properties of perfect and friendly numbers for some time; with the aid of computers, the list continues to grow.
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On October 4, 2009 at 9:40 am
Mathematics is unexplainable sometimes. I wrote one http://www.socyberty.com/Religion/Cool-The-Strangest-But-the-Most-Interesting-Numbers-Ever.442431
Thanks for your great post again.
On October 4, 2009 at 10:36 am
This is a great article. I particularly like number puzzles of any kind.
Christine
On October 4, 2009 at 10:52 am
Number puzzles are too confusing for me. I never did very well with math.
On October 4, 2009 at 9:05 pm
I can see how civilization could produce the leisure to devote to this pursuit. Can you imagine spending a lifetime pursuing perfect or friendly numbers? I wonder what kind of person would devote themselves to such a pursuit?
On October 4, 2009 at 10:34 pm
Another fantastic article! Very interesting. Numbers are great (:
Thank you for sharing!
On October 5, 2009 at 1:11 am
that was a good list, my friend, even if I’m a Math professor, I find those puzzle hard to finished.
On October 5, 2009 at 1:16 am
Absolutely amazing!..Great article and very interesting post about numbers and ancient mathematicians..I really enjoyed it! Thanks for sharing. Keep it up Mr Ghaz
On October 5, 2009 at 1:34 am
Another brilliant article from you Mr. ghaz..
Well researched article and educational too..Thanks a lot.
On October 5, 2009 at 1:55 am
Educative and interesting and novel.Enjoyed it.
On October 5, 2009 at 4:20 am
mathmatical equations–there is something strange about how numbers rule the world–very interesting post!!
On October 6, 2009 at 2:18 pm
It’s good to know the greatest minds also spend time playing computer games….
Interesting article, sir, I enjoyed it.
On October 6, 2009 at 4:59 pm
you’d think there would be a formula for predicting the next perfect number – what am I doing for the next couple of hours!?