Difference between revisions of "1990 OIM Problems/Problem 3"
(Created page with "== Problem == Let <math>f(x) = (x + b)^22-c</math>, be a polynomial with <math>b</math> and <math>c</math> as integers. a. If <math>p</math> is a prime number such that <math...") |
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== Problem == | == Problem == | ||
| − | Let <math>f(x) = (x + b)^ | + | Let <math>f(x) = (x + b)^2-c</math>, be a polynomial with <math>b</math> and <math>c</math> as integers. |
a. If <math>p</math> is a prime number such that <math>p</math> divides <math>c</math> and <math>p^2</math> does not divide <math>c</math>, show that, whatever the integer <math>n</math> is, <math>p^22</math> does not divide <math>f(n)</math>. | a. If <math>p</math> is a prime number such that <math>p</math> divides <math>c</math> and <math>p^2</math> does not divide <math>c</math>, show that, whatever the integer <math>n</math> is, <math>p^22</math> does not divide <math>f(n)</math>. | ||
Latest revision as of 00:22, 23 December 2023
Problem
Let 
, be a polynomial with 
and 
as integers.
a. If 
is a prime number such that divides and 
does not divide , show that, whatever the integer 
is, 
does not divide 
.
b. Let 
be a prime number other than 2, that divides . If divides for some integer , show that for every positive integer 
there exists an integer 
such that 
divides 
.
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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