20 Mind-Blowing Facts of Math’s Oldest Unsolved Problem (Do Odd Perfet Nos. Exist?)
Well, how ancient is math's oldest unsolved problem? Mathematically speaking, the oldest unresolved issue is 2000 years old. The best mathematicians have attempted to solve it but have yet to succeed. Italian mathematician Piergiorgio Odifreddi ranked it among the top four unsolved issues 2000. Finding a solitary number might be the answer to this puzzle. Therefore, mathematicians have verified numbers up to 102,200 using computers but failed to succeed. Why do you believe so many mathematicians are fascinated by this problem? - It is delightful, simple, and ancient. What more could you possibly want? Thus, this is Math's oldest unsolved problem: Are there any odd perfect numbers?
First, we must know what we mean when we say a perfect number. A positive integer equal to the sum of its appropriate divisors is called a perfect number. The result is equivalent to the number itself if all the components of an ideal number are added up, excluding the number itself. It's like discovering a number that enhances its grandeur because it likes itself so much.
If one utilizes the number six to illustrate, one may divide it (six) by 1, 2, 3, and 6. However, since 6 is the number, we can disregard it and remain with the appropriate divisors. They sum up to six, the number itself, when added together. It is why such numbers are said to be perfect. It may also be attempted with different numbers, such as 20. The appropriate divisors of 20 are 1, 2, 4, 5, and 10. Upon adding them up, the result is 22. Thus, twenty is not an ideal number. Doing this again for every other number will show that most overshoot or undershoot. Only 6 and 28 are perfect numbers between 1 and 100. If one ascends to 10,000, the following two perfect numbers are 496 and 8,128. Here are some of the most exciting things about perfect numbers and the obsession to find odd perfect numbers:
1. The first incredible fact is that the four perfect numbers known to ancient Greeks as early as 300 BC formed a pattern. One thing to notice is that each subsequent perfect number is one digit longer than the number presented before it. Therefore, these numbers are all just the sum of consecutive numbers, and you can think of each additional number as adding a new layer.
2. The last digit of the first four perfect numbers intersperses 6 and 8, meaning they are all even. These were the only perfect numbers known by the ancient Greeks and would be the only known ones for over a millennium. Since then, mathematical scholars have done research and, as of the publication of this article, have discovered 51 perfect numbers – all of which are even numbers.
3. Every perfect number bar for six is the sum of consecutive odd cubes of prime numbers. Subsequently, 28 is 13 + 33. 496 is equal to 13 + 33 +53 + 73. And 8,128 is equal to 13 + 33 +53 + 73 +93 + 153.
4. Another mind-blowing thing is that if you inscribe the perfect numbers in binary form, 610 = 1102, and 2810 = 111002. 49610 = 1111100002. And 8,12810 = 11111110000002. It is also a string of ones (in the pattern 2, 3, 5, 7 . . .. ones) followed by a series of zeros (in the pattern 1, 2, 4, 6 . . ..). Thus, if one carves them out, they equal consecutive powers of two.
5. The obsession with number perfection led to significant developments in the math field, notably with the introduction of Euclid’s formula that gives a perfect number as (2P – 1) x (2P – 1) as long as this value is prime. In half a shake, since one multiplies the value by (2P – 1) – an even value - it will always result in an even number. Euclid had found a formula to produce even perfect numbers, but he did not prove this was the only methodology. Accordingly, mathematicians hypothesized that there could be other approaches to obtaining perfect numbers plus odd ones. Centuries later, the Greek philosopher Nicomachus published Introduction to Arithmetic (Arithmetike eisagoge), the standard arithmetic text that would guide the succeeding millennium. In it, he put forward five conjectures that he believed to be true but did not go to the trouble of attempting to prove them. His first conjecture was that the nth perfect number has n digits. Second, all perfect numbers are even. Third, all perfect numbers end in 6 and 8 interchangeably. Fourth, the Euclid algorithm constructs every even perfect number. And five, there are infinitely many perfect numbers.
6. French polymath Marin Mersenne extensively studied numbers that took this form. In 1644, he published in a book a list of 11 values of P, for which he averred that they corresponded to primes. Numbers for which this is true are now known as Mersenne Primes. In Mersenne’s list, the first seven exponents of P generate primes, consistent with the first seven perfect numbers. However, for some of the more significant numbers, e.g. (267 – 1), Mersenne admitted not verifying whether they were prime.
7. In 1732, Leonhard Euler made three fundamental discoveries: finding the eighth perfect number and verifying that (231 – 1) was prime. As Mersenne had forecast, he invented a new armament, the Sigma function, for his other two discoveries. The function aims to take all the divisors of a number, including the number itself, and add them up. If a mathematician took any number, say 28, and summed up all its divisors, the outcome is 56, double the number we started with. It is a discovery that he showed to apply and be valid for all perfect numbers. Consequently, it was deduced that the Sigma function of a perfect number will at all times result in double the number itself since the Sigma function incorporates the number as one of its divisors. Crucially, Euler proved that every perfect number has Euclid’s form.
8. There are famous quotes associated with this math problem. Mersenne is believed to have said, "To tell if a given number of 15 to 20 digits is prime or not all time would not suffice for the test." Math historian William Dunham called the Euclid-Euler theorem "the greatest mathematical collaboration in history." English mathematician Peter Barlow penned that Euler's eighth perfect number "Is the greatest that ever will be discovered for as they are merely curious without being useful, it is not likely that any person will ever attempt to find one beyond it." Descartes argued, "As for me, I judge that one can find real odd perfect numbers. But whatever method you use, it takes a long time to look for these."
9. In 1874, Edouard Lucas demonstrated that (267 – 1) was not prime, even if he could not present its factors. In 1901, Frank Nelson Cole lectured at the American Mathematical Society. During the presentation, he quietly and without warning walked to one side of the blackboard and engraved down (267 – 1) = 147,573,952,589,676,412,927. He then strolled to the other side of the blackboard and multiplied 193,707,721 x 761,838,257,287, providing a similar answer. When he finally sat down, his entire audience applauded. He later confessed that it took him three years of working on Sundays to solve this problem. Shockingly, modern computers could solve this problem in < a second.
10. From 500 BC to 1952, mathematicians discovered just 12 Mersenne primes and only 12 perfect numbers. The main issue was verifying if massive Mersenne numbers were in point-of-fact prime. Nonetheless, in 1952, American mathematician Raphael Robinson wrote a computer program to solve this problem, and he ran it on the fastest computer of the time, the SWAC. Within 10 months, he generated the successive five Mersenne primes and, thus, equivalent perfect numbers.
11. By the turn of the millennium, new Mersenne primes had been rapidly discovered, all using computers. The largest Mersenne prime at the end of 1952 was (22,281 – 1), which is 687 digits long. By the end of 1994, the largest Mersenne prime was (2859,433 – 1), which is 258,716 digits long.
12. Since these numbers were developing to be exceedingly massive, finding various end primes became gradually tricky, even for supercomputers. Consequently, in 1996, computer scientist George Woltman unveiled the Great Internet Mersenne Prime Search or GIMPS. GIMPS is a form of cloud computing network that distributes the work over many computers allowing curious persons to volunteer their computer power to aid the search for Mersenne primes. The project has been vastly fruitful, enabling the discovery of 17 new Mersenne primes, 15 of which were the largest known primes then.
13. Participating in this program can give you bragging rights and international accolades. If a participant’s computer in the GIMPS distributed resource-sharing network discovers a new Mersenne prime, the person is listed as its discoverer and added to a list that includes some of the best and most famous mathematicians of all time.
14. In the year 2017, Church Deacon John Pace discovered the 50th Mersenne Prime after being a participant in the GIMPS program. The number (277,232,917 – 1) is >23 million digits long and was also the largest known prime then. After celebrating this achievement, the Japanese publishing house Nanairosha published the book, “The Largest Prime Number of 2017.” The book is the whole number spread over 719 splendid pages. The book swiftly soared to the first-selling on Amazon and sold out in four days.
15. In 2018, a year later, the 51st Mersenne Prime was discovered. Its value is (282,589,933 – 1), comprising 24,000,860 2048 digits. Its corresponding perfect number has 49 million digits. The 51st Mersenne Prime was also published in another sold-out Amazon publication.
16. Numbers that are very close to becoming odd perfect numbers are termed spoofs. Spoofs are a higher set of numbers. Thus, odd perfect numbers have all the properties of spoofs and supplementary characteristics. In 2022, Pace Nielsen and a Bringham Young University (BYU) team discovered 21 spoof numbers, including a Descartes number. While this research team discovered some new properties of spoofs, they failed to discover any that exclude odd perfect numbers.
17. Many math scholars and curious minds continue to obsess with the discovery of these perfect numbers. For instance, a group of mathematicians called the “Perfect Club” has dedicated substantial research to discover new perfect numbers.
18. There have been various false, misleading, incorrect, or unproven solutions and conjectures to this problem. In the 13th century, Egyptian mathematician Ibn Fallus published a list comprising 10 perfect numbers and their corresponding values of P. Three of these perfect numbers were disproved as perfect, whereas the remaining ones were proven perfect. In the 19th century, mathematician Édouard Lucas claimed to have found an odd perfect number, but it turned out to be a hoax. The fifth perfect number is eight digits long, which disproved Nicomachus’s first conjecture. Also, one notices that the fifth and sixth perfect numbers' last digits are 6 for each. As a result, that disproved Nicomachus’s third conjecture that all perfect numbers end in a 6 or 8 alternately. In addition, the ABC Conjecture, proposed by Japanese mathematician Shinichi Mochizuki, states that there are only finitely many odd perfect numbers. However, Mochizuki’s conjecture is still being deliberated, and there is no proof of this article's publication.
19. Getting involved in solving this problem can earn you monetary rewards. In the 1950s, mathematician Paul Erdős offered a $500 prize for anyone who could find an odd perfect number. To this day, the prize remains unclaimed. There is also a $250,000 prize for the computer user whose machine registers the first billion-digit prime via the GIMPS program.
20. As of right now, this problem has no real-world applicability. Why would someone solve a math problem without considering the applications? In any case, the only way to learn is to solve the presented problem. To predict the result in advance, the scholar has to wait. Mathematicians may solve the problem without practical significance, but it might still be helpful. Think of the number theory, which had yet to be practical use for almost two millennia. It was merely mathematicians pursuing their curiosity, figuring out intriguing puzzles, establishing a foundation of meaningless mathematics, and proving one result after another. Later on in the 20th-century digital revolution, researchers discovered they could use this basis as the foundation for cryptography. People unknowingly need this knowledge for everything, even government secrets and SMS messages for safeguard. While we may never know what potential impact these perfect numbers may have in the future, we know that when mathematicians face a challenge, they approach it head-on, generate fresh solutions, and finally, helpful something emerges from that process.
When all is said and done, Discovering Odd Perfect Numbers remains the oldest unsolved problem in math. Euler was right when he said whether there are any odd perfect numbers is a most challenging question. Still, who does not love perfection? It is like finding a unicorn or a pot of gold at the end of a rainbow - elusive and magical. And what’s more impressive than a perfect number? It is like the holy grail of math. But mathematicians have an obsession with finding perfect odd numbers. Will mathematicians ever come close to finding a perfect odd number? They've been chasing after it for centuries, and this trend will likely persist for decades, if not centuries, to come. It’s like a never-ending maze with the end out of sight entirely!
Hence, here’s to all the curious mathematicians who still obsess with number perfection. May your quest to find the perfect odd number succeed so you can finally put this obsession to rest. But then again, if we are realistic, we all know these supercharged brains will move on to the following mathematical mystery to solve and become obsessed! It is a vicious cycle, but at least these brain math nerds are having fun while at it. I love the mathematicians as they continue to explore the mysteries of mathematics and push the boundaries of human knowledge. My key takeaway from the perfect number obsession is that math is not just about finding answers; it's also about the journey, the challenge, and the beauty of the numbers themselves.
