Some questions seem to call for an equation to be solved but lead to unrelated questions, such as purely arithmetic problems concerning prime numbers, for example.
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The prime numbers are the numbers divisible exactly by two numbers: 1 and themselves. Thus, the smallest prime numbers are 2, 3, 5, 7, 11, 13, etc. There are thus an infinity of them and they are at the origin of very difficult questions, unresolved at present and probably for a long time to come, such as the conjecture of Goldbach, which dates from 1742, according to which any even number (at from 4) would be the sum of two prime numbers (for example 4 = 2 + 2; 6 = 3 + 3; 8 = 5 + 3… 20 = 13 + 7; etc.) conjecture probably true but still unproven .
The essential property of prime numbers is simpler. Here it is: any whole (non-prime) number can be written in a unique way as a product of prime numbers.
So, 10 = 2 x 5; 506 = 2 x 11 x 23. To perform such a decomposition by hand, one can try dividing the given number by the prime numbers in order from the smallest: 2, 3, 5, 7, etc. You can also use a software computer algebra (there are free ones). This leads to a mathematical-historical question which may seem very far removed from prime numbers and recalls the famous problem that Gustave Flaubert sent to his sister where he asked her captain’s age without any of the data of the statement being able to lead to a solution. Here’s a question that might sound similar:
Question:
The last day of a month of the first World War, a shell put an end to the life of a young captain. The same day, in a neighboring country, a peasant unearthed a soldier who had died in a great battle of yesteryear. The date of the month, multiplied by the length in feet of the partisan, multiplied by half the age of the captain, multiplied by a quarter of the time (in years) which separated these two deaths is equal to 225.533. We ask the captain’s age.
Responnse :
22 years old. This strangely worded problem consists in factoring 225.533, which is laborious by hand since the smallest factor is 7. We first find: 225.533 = 7 x 32.219. The second factor is 11: 32.219 = 11 x 2.929. The divisor 101 is then highlighted: 2.929 = 29 x 101 hence the factorization:
225.533 = 7x11x29x101
The last day of the month can therefore only be the 29th, which implies that the year is a leap year, therefore 1916, the only leap year of the Great War. It’s time to find out what a partisan is. This is a spear used by a foot soldier in the 15thand and XVIand centuries. It measures between 2 and 4 meters. By evaluating the foot at 30 cm, its length is between 6 and 12 feet. Given the statement, it is therefore worth 7 or 11 feet. The quarter of the time between the two deaths can only be 101 and the age of the captain, which cannot be 14, is 22 and the partisan is 7 feet tall. The sustainer died in 1512 so, probably at the Battle of Ravenna.
Learn more about Hervé Lehning
Normalien and agrégation in mathematics, Hervé Lehning taught his discipline for a good forty years. Crazy about cryptography, member of the Association of encryption and information security reservists, he has in particular pierced the secrets of Henri II’s cipher box.
- His blog MATH’WORLD on Futura
- the latest book by Hervé Lehning :
Also to discover: The universe of secret codes from Antiquity to the Internet published in 2012 by Ixelles.
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