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Old 2018-01-19, 04:51   #235
gophne
 
Feb 2017

3×5×11 Posts
Default Lucky Polynomials

Hi Everybody

We know that the polynomials of Euler/Legendre x^2+x+41 and x^2-x+41 generate the series of primes called Euler numbers;

41 43 47 53 61 71 83 97 113 131 151 173 197 223 251 281 313 347 383 421 461 503 547 593 641 691 743 797 853 911 971 1033 1097 1163 1231 1301 1373 1447 1523 1601

for the first 40 terms.

Question, is it also commonly known that these values are also apparently generated by the polynomials;

x^2+3x+43 and x^2-3x+43
x^2+5x+47 and x^2-5x+47
x^2+7x+53 and x^2-7x+53........

That is each new polynomial being
f(x)=x^2+ x+41 plus 2x+2,
f(x)=x^2+3x+43 plus 2x+4
f(x)=x^2+5x+47 plus 2x+6
f(x)-x^2+7x+53 plus 2x+8.......

At least for the next +-36 polynomials as well.

i could not find these "derived' polynomials mentioned anywhere, except the Euler/Legendre forms of the polynomial.

Caveat: Not sure if similar derived polynomials have been published/discussed else where. Not encountered in my searches for same in Wikipedia.

Last fiddled with by gophne on 2018-01-19 at 04:55 Reason: Correction to number of polynomials generating Euler numbers
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