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For Theatregirl -- math problem
Is the expression x² + x + 41 prime for all non-negative integers x?
0² + 0 + 41 = 41 (prime)
1² + 1 + 41 = 43 (prime)
2² + 2 + 41 = 47 (prime)
3² + 3 + 41 = 53 (prime)
4² + 4 + 41 = 61 (prime)
5² + 5 + 41 = 71 (prime)
Note: this sequence can also be obtained by starting with 41, then add on 2, then add on 4, then 6, then 8,...
Extra credit (+3 GOE): Can you settle this question without a calculator?
seems to be all prime. if you look at the difference between the results in the numbers, it is all even intergers of 2, 4, 6, 8 etc
out of curosity, where do you get stuff like this? are you a math teacher?
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Yes, I am a mathematics professor.
This problem was invented by the prolific Swiss mathematician Leonard Euer in the eighteenth century. There was quite a bit of interest in those days in studying the distribution of primes among the integers. People tried to invent polynomials that generated all primes, and Euler's try, with the polynomial x² + x + 41, was one of the simplest.
What about 25² + 25 + 41? Still prime? How about 38² + 38 + 41?
In 1752 Goldbach proved that no polynomial generates all primes (in particular, this one doesn't). What number should you try to get an answer that is not prime?
BTW, want to make a million dollars? That's the prize offered for anyone who can prove the Goldbach Conjecture: Every even number greater that 4 can be written as the sum of two primes.
How high do you have to go? Does infinity have to be in the equation? Who's offering the prize? It could pay for my dissertation research!
Originally Posted by Mathman
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