![]() ![]() Paul Jenkins, also at BYU, proved in 2003 that the largest prime factor of an OPN must exceed 10,000,000. Norton proved that if an OPN is not divisible by 3, 5 or 7, it must have at least 27 prime factors. In 1888, for instance, James Sylvester proved that no OPN could be divisible by 105. These were neither the first nor the last restrictions established for the hypothetical OPNs. ![]() He showed not only that the number of OPNs with k distinct prime factors is finite, as had been established by Leonard Dickson in 1913, but that the size of the number must be smaller than 2 4 k. “That motivated me to study number theory in college and try to move things forward.” His first paper on OPNs, published in 2003, placed further restrictions on these hypothetical numbers. “Seeing that progress could be made on this problem gave me hope, in my naiveté, that maybe I could do something,” Nielsen said. He delved into the literature, coming across a 1974 paper by Carl Pomerance, a mathematician now at Dartmouth College, which proved that any OPN must have at least seven distinct prime factors. Nielsen first learned about perfect numbers during a high school math competition. It involves the closest thing to OPNs yet discovered. But last June he hit upon a new way of approaching the problem that might lead to more progress. And, also like his peers, he does not believe a proof is within immediate reach. Like many of his 21st-century peers, Nielsen thinks there probably aren’t any OPNs. Nielsen, now a professor at Brigham Young University (BYU), was ensnared by a related question: Do any odd perfect numbers (OPNs) exist? The Greek mathematician Nicomachus declared around 100 CE that all perfect numbers must be even, but no one has ever proved that claim. Euler proved 2,000 years later that this formula actually generates every even perfect number, though it is still unknown whether the set of even perfect numbers is finite or infinite. For example, if p is 2, the formula gives you 2 1 × (2 2 − 1) or 6, and if p is 3, you get 2 2 × (2 3 − 1) or 28 - the first two perfect numbers. He showed that if p and 2 p − 1 are prime numbers (whose only divisors are 1 and themselves), then 2 p −1 × (2 p − 1) is a perfect number. The updates got out of control and I stopped documenting them here.But Pythagoras was aware of perfect numbers back in 500 BCE, and two centuries later Euclid devised a formula for generating even perfect numbers. Button to toggle visibility of completed items.Start of Item Walk-through (thanks /u/ScottishWolverine).Completed items are now feature a strike-through.Upgrade to Boostrap 3 and further HTML updates/optimizations, which fixed the Mobile Profile issues.Many more great additions from the community. A search mechanism has been added thanks to sethxd. ![]()
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