example of gcd
If one calculates values of the polynomial
. . .
It is indeed true. The reason of this fact is not particularly deep. It is easily understood if we can factorize this polynomial (http://planetmath.org/GroupingMethodForFactoringPolynomials):
and the next value is
Thus and have as their common factor at least the number , which is .
Moreover, we may show that the greatest common divisor of and is except in the case where it is .
Let be a common divisor, greater than 1, of the first factors and of (1) and (2). It’s clear that is odd (http://planetmath.org/OddNumber). Then must divide the difference and the sum and hence also the difference . It means that . Thus the only possible common prime factor of and is 7. If we denote where , we see that
Note. The sequence formed by the successive values of is number A111002 in Sloane’s register (https://oeis.org/A111002http://www.research.att.com/ njas/sequences/).
|Title||example of gcd|
|Date of creation||2014-10-25 7:09:30|
|Last modified on||2014-10-25 7:09:30|
|Last modified by||pahio (2872)|