For thousands of years mathematicians have been interested in the question of the existence of odd perfect numbers. In the process of studying it, they compiled an incredible list of restrictions for these hypothetical objects. But new ideas on this score may appear due to the study of other objects close to them.
If odd perfect numbers exist, they will have to satisfy an absurdly long list of constraints.
As a high school student, Pace Nielsen faced a math question in the mid-90s that he still struggles with to this day. But he is not upset: the problem that fascinated him, the hypothesis of odd perfect numbers, has remained open for more than 2,000 years, making it one of the oldest unsolved problems in mathematics.
Part of that long-lived charm comes from the simplicity of the wording. A number is called perfect if it is a positive integer, n, whose divisors add up to twice the number, 2n. The first and simplest example is 6, whose divisors 1, 2, 3, and 6 add up to 12, or 2 * 6. Then comes 28, with divisors of 1, 2, 4, 7, 14 and 28, giving a total of 56. The next examples are 496 and 8128.
Leonard Euler formalized this definition in the 18th century, introducing his sigma function, which denotes the sum of the divisors of a number. Thus, for perfect numbers, σ (n) = 2n.
Leonard Euler formulated many formal rules regarding working with perfect numbers.However
, Pythagoras knew about perfect numbers as early as 500 BC, and two centuries later Euclid derived a formula for obtaining even perfect numbers. He showed that if p and 2 p - 1 are prime numbers (divisors of which are only 1 and this number itself), then 2 p − 1 * (2 p - 1) will be a perfect number. For example, if p = 2, then the formula gives 2 1 * (2 2 - 1), or 6. If p = 3, then the formula gives 2 2 * (23 - 1), or 28 - the first two perfect numbers. 2000 years later, Euler proved that this formula produces all even perfect numbers, although it is still unknown whether the set of perfect numbers is finite or infinite.
Nielsen, now a professor at Brigham Young University, got carried away with a related question: Do odd perfect numbers exist? Greek mathematician Nicomachus of Gerasa circa AD 100 stated that all perfect numbers must be even, but no one has proven this statement.
Like many of his colleagues from the 21st century, Nielsen believes that there are not very many perfect numbers. And, together with them, he believes that the proof of this hypothesis will not be obtained soon. However, in June he came acrossa new approach to this task, perhaps capable of taking it further. And it is associated with the object closest to odd perfect numbers from all so far discovered.
Shrinking web
Nielsen first learned about perfect numbers in a math competition at school. He delved deeper into literature, and stumbled upon the 1974 work of Karl Pomeranz , a mathematician now serving at Dartmouth College. He proved that any odd perfect number must have at least seven different prime factors.
“I in my naivety decided that I can do something in this area, if progress is possible at all,” said Nielsen. "It inspired me to study number theory in college and try to make progress." His first work on odd perfect numbers, published in 2003, placed additional constraints on these hypothetical numbers. He showedthat not only the number of odd perfect numbers with k different prime divisors is finite, as Leonard Dixon proved in 1913, but also that the size of this number should not exceed 2 4 k .
And this was not the first nor the last limitation imposed on hypothetical odd perfect numbers. For example, in 1888, James Sylvester proved that an odd perfect number cannot be divisible by 105. In 1960, Carl K. Norton proved that if an odd perfect number is not divisible by 3, 5, or 7, it must have at least 27 prime factors. Paul Jenkins in 2003 provedThat the largest prime divisor of odd perfect number must be greater than 10 000 000. Pascal ochem and Mihaol Rao then found that odd perfect number must be greater than 10 1500 , and then pushed the boundary to 10 2000 . Nielsen showed in 2015 that an odd perfect number must have at least 10 different prime factors.
Pace Nielsen, mathematician at Brigham Young University
Even in the 19th century, the number of restrictions was such that Sylvester concluded that "the emergence of an odd perfect number - a sort of escape from the complex network of conditions surrounding it on all sides - would be almost a miracle." After more than a hundred years of such a development of events, the existence of such numbers raises even more doubts.
“Proving the existence of something is easy if you can find just one example,” said Jon Voight , a mathematics professor at Dartmouth. "But proving that something doesn't exist can be very difficult."
The main approach until now has been to compare all conditions that limit odd perfect numbers in order to find out if any pair of them is incompatible - that is, that no number can satisfy both constraints at once. “The patchwork of conditions we have obtained to date makes the odd perfect numbers extremely unlikely,” Voight said, echoing Sylvester. "And Pace has been adding new items to this list for many years."
Unfortunately, no incompatible properties have yet been found. Therefore, in addition to additional restrictions on odd perfect numbers, mathematicians will probably need new strategies.
To this end, Nielsen is already considering a new attack plan based on a common tactic of mathematicians: the study of many numbers through the study of their close relatives. In the absence of odd perfect numbers suitable for direct study, he and the team study "imitations" of odd perfect numbers, which are very similar to the real ones, but have some interesting differences.
Understanding perfect numbers
- . σ(n) = 2n, .
:
σ(20) = 1 + 2 + 4 + 5 + 10 + 20 = 42; 2 * 20 ≠ 42, 20 – .
σ(28) = 1 + 2 + 4 + 7 + 14 + 28 = 56; 2 * 28 = 56, 28 – .
1. σ(a × b) = σ(a) × σ (b) , , a b – .
2. σ(pa) = 1 + p + p2 + … + pa p a.
:
σ(20) = σ(22 × 5) = σ(22) × σ(5) [ ] = (1 + 2 + 22)(1+5) [ ] = 42
σ (28) = σ (2 2 × 7) = σ (2 2 ) × σ (7) [by the first rule] = (1 + 2 + 2 2 ) (1 + 7) [by the second rule] = 56
New seductive misses
The first imitation of an odd perfect number was found in 1638 by Rene Descartes - and he was one of the first outstanding mathematicians who considered the existence of odd perfect numbers possible. “I believe Descartes was trying to find odd perfect numbers, and his calculations led him to the first imitation,” said William Banks , a number theorist at the University of Missouri. Apparently, Descartes hoped that the number he created could be changed to obtain a real odd perfect number.
But before diving into Cartesian imitation, it is helpful to understand a little about how mathematicians describe perfect numbers. Euclid's time theorem states that any integer greater than 1 can be expressed as a product of prime numbers raised to certain powers. For example, 1260 can be factorized like this: 1260 = 2 2 × 3 2 × 5 1 × 7 1 , and not list all 36 factors separately.
Once a number takes this form, it becomes much easier to compute the Euler sigma function that sums its divisors, thanks to two formulas that Euler also proved. First, he demonstrated that σ (a × b) = σ (a) × σ (b) if and only if a and b are coprime - that is, they have no common prime divisors. For example, the numbers 14 (2 × 7) and 15 (3 × 5) are relatively prime. Second, he showed that for any prime number p in a positive integer degree a, σ (p a ) = 1 + p + p 2 +… + p a .
Returning to our previous example, σ (1 260) = σ (2 2 × 3 2 × 5 1 × 7 1 ) = σ (2 2 ) × σ (3 2) × σ (5 1 ) × σ (7 1 ) = (1 + 2 + 2 2 ) (1 + 3 + 3 2 ) (1 + 5) (1 + 7) = 4 368. Note that in this case σ (n) is not equal to 2n, which means that 1260 is not a perfect number.
Rene Descartes found the first imitation of a perfect number
Now we can parse the Cartesian imitation - the number 198 585 576 189, or 3 2 × 7 2 × 11 2 × 13 2 × 22 021 1 . Repeating the above calculations, we find that σ (198 585 576 189) = σ (3 2 × 7 2 × 11 2 × 13 2 × 22.021 1 ) = (1 + 3 + 32 ) (1 + 7 + 7 2 ) (1 + 11 + 11 2 ) (1 + 13 + 13 2 ) (1 + 22.021 1 ) = 397 171 152 378. And this is equal to twice the original number, which means that it should be a real perfect number - only the number 22,021 is not prime.
Therefore, this number of Descartes is an imitation. If we pretend that 22,021 is prime and apply Euler's rules to the sigma function, Descartes's number behaves like a perfect number. However, 22 021 is actually the product of 19 2 and 61. If we could write Descartes's number correctly as 3 2 × 7 2 × 11 2 × 13 2 × 19 2 × 61 1, then σ (n) would not equal 2n. Weakening some of the rules, we get a number that seems to satisfy our requirements - this is the essence of imitation.
It took 361 years to discover the second imitation number of an odd perfect number. Voight did this in 1999, and published the discovery four years later. Why so long? “Finding an imitation number is like finding an odd perfect number; both are similarly arithmetically complex, ”Banks said. And their search was not a priority for mathematicians. However, Voight was inspired by an excerpt from Richard Guy's Unsolved Problems in Number Theory, where he wrote about the search for new imitations. Voight tried, and eventually found a new imitation, 3 4 × 7 2 × 11 2 × 19 2× (−127) 1 , or −22 017 975 903.
Unlike Descartes' example, here all divisors are prime, but one of them is negative - therefore this number is an imitation, not a true odd perfect number.
Simulate odd perfect numbers
:
198 585 576 189, 32 × 72 × 112 × 132 × 22 0211.
-: σ(198 585 576 189) = σ(32 × 72 × 112 × 132 × 22,0211) = (1 + 3 + 32)(1 + 7 + 72)(1 + 11 + 112)(1 + 13 + 132)(1 + 22,0211) = 397 171 152 378 = 2 × 198 585 576 189.
22 021 , 192 × 61. .
:
−22 017 975 903, 34 × 72 × 112 × 192 × (−127)1.
-: σ(−22 017 975 903) = σ(34 × 74 × 112 × 192 × (-127)1) = (1 + 3 + 32 + 33 + 34)(1 + 7 + 72)(1 + 11 + 112)(1 + 19 + 192)(1 + (-127)1) = -44 035 951 806 = 2 × −22 017 975 903
-127 – , – .
After Voight held a seminar at Brigham Young University in December 2016, he discussed this number with Nielsen, Jenkins and others. Shortly thereafter, the university team set out on a systematic computational search for other imitations. They would choose the smallest bases and exponents, like 3 2 , and then computers combed through variants of additional bases and exponents that would simulate a perfect number. Nielsen decided that this project would simply be a stimulating research experience for his students, but the results of the analysis exceeded his expectations.
Sifting through the possibilities
After running 20 processors continuously for three years, the team discovered every possible imitation of a perfect number that could be written using six bases or less - 21 in total, including examples from Descartes and Voight - and two more simulations with seven divisors. Searching for simulations with a large number of dividers on computers was impractical and too time consuming. Nevertheless, the group has collected enough examples to discover previously unknown properties of imitations.
The group found that for any given number of bases k, there is a finite number of imitations, which coincides with Dixon's 1913 result for true odd perfect numbers. “However, if k goes to infinity, the number of imitations also becomes infinite,” said Nielsen. This was unexpected, he added, given that starting this project, he was not sure of discovering even one new odd imitation, let alone show that their number is infinite.
Another surprise stemming from a result first proved by Euler: all prime bases of an odd perfect number, except one, must have even degrees. One must have an odd degree - this is called the Euler degree. Most mathematicians believe that the Euler degree for odd perfect numbers is always 1, but the team has shown that simulations can have as large as they like.
The team found some of the findings by weakening the requirements in the definition of imitation, since there are no clear mathematical rules for describing them - only that they must satisfy the equality σ (n) = 2n. The researchers allowed the existence of non-prime bases (as in Descartes' example) and negative bases (as in Voight's example). However, they went further by allowing imitations to have several of the same bases. One radix, for example, might be 7 2 and the other 7 3 , and they are written separately, and not as 7 5 . Or they let the reasons repeat themselves, as in the imitation 3 2 × 7 2 × 7 2 × 13 1 × (−19) 2... The term 7 2 × 7 2 can be written as 7 4 , but then the simulation will fail, because the expansion of the parentheses in the modified sigma function would be different.
Given the significant difference between imitations and real odd perfect numbers, one might ask the question: how do the former help in finding the latter?
The way forward?
Nielsen said that imitations are generalizations of odd perfect numbers. Odd perfect numbers are a subset within a larger family, which includes imitations, so odd perfect numbers must have all the properties of imitations, as well as additional, even more stringent restrictions (like, for example, the condition that all grounds must be simple) ...
“Any behavior of the larger set must be followed for the smaller subset,” Nielsen said. "So if we find imitation behavior that doesn't apply to a more limited class, we can automatically discard the possibility of odd perfect numbers." If, for example, it can be shown that all simulations are divisible by 105 - which is impossible for odd perfect numbers, as Sylvester showed in 1888 - then the problem will be solved.
So far, however, they have not succeeded. "We have discovered new facts about imitations, but none of them deny the existence of odd perfect numbers," said Nielsen, "although this possibility still remains." By further analyzing the currently known imitations, and, possibly, supplementing their list in the future, Nielsen (and both these directions are developing thanks to him) and other mathematicians can discover new properties of imitations.
Banks finds this approach worthwhile. “Exploring odd imitations can be helpful in understanding the structure of odd perfect numbers, if any,” he said. "And if there are no odd perfect numbers, studying odd imitations can lead to proof of this."
Other experts on odd perfect numbers are not so optimistic. The team at Brigham Young University “did a great job,” Voight said, “but I'm not sure we're getting close to attacking the odd perfect number problem. This is truly a task for the ages, and it is likely that it will remain so. "
Paul Pollack , a mathematician at the University of Georgia, is also cautious: “It would be cool if we could look at the list of imitations and see some of their properties, and somehow prove that odd perfect numbers with this property do not exist. It would be just a dream, but it seems too good to be true. "
Nielsen agreed that there was little chance of success, but to solve this ancient problem, mathematicians must try everything. Moreover, the study of imitations is just beginning. His group has taken some early steps and has already discovered unexpected properties of these numbers. Therefore, he is optimistic about the possibility of discovering additional "hidden structures" within the imitations.
Nielsen has already identified one plausible tactic based on the fact that all imitations found to date, apart from Descartes' original example, have at least one negative basis. If we prove that all other imitations must have a negative base, then this proves that odd perfect numbers do not exist, since, by definition, their bases must be simple and positive.
“This seems like a more difficult task,” Nielsen said, as it touches on a larger and more general category of numbers. "But sometimes, when you transform a problem into a seemingly more difficult one, you can see the path to the solution."
In number theory, patience is required - sometimes the question is easy to ask but difficult to answer. “You have to think about the task, sometimes for a long time, and pay special attention to it,” said Nielsen. - We are moving forward. We are digging a mine. We hope that if we dig long enough we can find a diamond. "