By Natalia Kokoromyti
Christian Goldbach (1690-1764) Sometimes the most challenging problems in mathematics are the ones that can be phrased in the most seemingly simplistic way. In 1742, Prussian historian and mathematician Christian Goldbach speculated that every integer greater than 5 can be written as the sum of 3 primes, for example 21=11+7+3 (notice that a number greater than 1 is called prime, if its only positive divisors are itself and 1). After some time, Swiss mathematician Leonhard Euler formulated the strong version of the Goldbach conjecture, suspecting that all positive even integers greater than 2 can be expressed as the sum of two primes. Little did they know that two and a half centuries later, the London-based colossus in book publishing “Faber and Faber”, in an effort to boost the promotional campaign of one of its upcoming books titled “Uncle Petros and Goldbach’s Conjecture”, would offer a whopping 1 million dollars to whoever managed to prove the conjecture during the time between 20th of March 2000 to 20th of March 2002. Unfortunately no one succeeded in this daring endeavor and the conjecture remains unproven till this very day. However, in 2008, Tomás Oliveira e Silva, a researcher from the Universidade de Aveiro in Portugal, performed a distributed computer search and verified that the conjecture holds for as far as 12*10^17 (today this number has become 4*10^18). Evidently though, with this procedure we can never show that the conjecture holds for all such integers, hence mathematicians still hope for a solid proof of Goldbach’s speculation. In 1966, Chen Jing-Run, a Chinese mathematician, made significant strides forward when he proved that a sufficiently big even number can be expressed as the sum of a prime and a number that is the product of at most two primes (e.g.18=3+(3*5). Additionally, in 1995, French mathematician Olivier Ramaré showed that every even number greater than 2 is the sum of at most 6 primes. To conclude, this is where the mathematical community stands right now…but who knows what the future has in store for this much-discussed conjecture? *Definition of conjecture (From Merriam-Webster): a proposition (as in mathematics) before it has been proved or disproved.
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By Natalia Kokoromyti
Back in 2005 the German newspaper “Die Welt” (“The World”) decided to invite its readers to play an interesting and really promising game. It dared them to do the following: “Take a 3-digit number of your choice and write it down twice, hence forming a new integer. If for example your initial number was, let’s say, 761 your new number will be 761761. Now divide this 6-digit integer with 7 and the remainder of this division will be, from now on, your lucky number.” For someone that has somewhat of a close connection with the fundamentals of number theory it is quite intuitive that the possible remainders for when a number is divided by 7 are 0,1,2,3,4,5 and 6. So the newspaper invited its readers to write down on a postcard the lucky number that they got, and send it to the company’s headquarters in Berlin. Then the company would be obliged to send back to the reader a number of 100$ bills that is equal to their respective lucky number. If meanwhile, with your innate curiosity(!), you tried this out yourself I bet that the lucky number that you got is 0! Am I mentalist? Maybe, conceivably, possibly. Yet I have to admit that this observation should not be attributed to my mind-reading abilities that I frequently attempt to finesse, but to an even more brain-fuddling property of the integers. Because let me tell me you, you weren’t the only one whose lucky number was 0. Actually all the readers that took part in this problem got 0 as their lucky number! The explanation of this phenomenon lies in a hidden property of the integers. To be more specific, notice that writing a 3 digits number two times consecutively is equivalent to multiplying this 3-digit number with 1001. And because 1001 is a multiple of 7, so is the 6-digit number. Note that with the same reasoning we can use other divisors of 1001, like 11 or 13 instead of 7. |
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May 2022
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