The discussion revolves around the properties of the quadratic expression 20*x^2-1, particularly focusing on the nature of its integer iterations and the factors of the resulting numbers. Participants explore the last digits of these factors and their implications in number theory.
Participants express varying viewpoints on the nature of the factors and the applicability of modular arithmetic, indicating that multiple competing views remain. The discussion does not reach a consensus on the conclusions drawn from the exploration of the quadratic expression.
Some participants acknowledge the complexity of the problem and the limitations of their approaches, noting that ruling out specific primes as factors is straightforward, but generalizing these findings remains challenging.
Do you have any idea? Without the "-1", which is the last digit?Janosh89 said:Ian in London (South).Interests Number Theory. Prime Numbers...
Integer iterations of this quadratic expression only yield decimal numbers whose factors end with the digit _1, or _9. Why?
It is not so trivial if you want to consider all factors of the number. 13*23 can give the correct last digit, for example.fresh_42 said:Do you have any idea? Without the "-1", which is the last digit?
The factors are in the set 11,19,29,31,41,59,61,71 and so on but can include prime numbers (^2), (^4),(^6)...fresh_42 said:Do you have any idea? Without the "-1", which is the last digit?
299 (13*23) is not expressible by 20*x^2-1 ? where x is an integermfb said:It is not so trivial if you want to consider all factors of the number. 13*23 can give the correct last digit, for example.
@Janosh89: Can you rule out some factors based on modular arithmetic?
Can you generalize that?
Yes, I erroneously thought it was about the digits of ##20x^2-1## itself.mfb said:It is not so trivial if you want to consider all factors of the number.
mfb said:It is not so trivial if you want to consider all factors of the number. 13*23 can give the correct last digit, for example.
@Janosh89: Can you rule out some factors based on modular arithmetic?
Can you generalize that?
fresh_42 said:Yes, I erroneously thought it was about the digits of ##20x^2-1## itself.
Interesting question though and indeed not trivial.
Hope you got my reply. Thanks very much for responce to my initial question.fresh_42 said:Yes, I erroneously thought it was about the digits of ##20x^2-1## itself.
Interesting question though and indeed not trivial.
I don't understand what this means. Is it another question, or an attempt at answering the original question?Janosh89 said:This is applicable to a sub-class of prime pairs, 59;61 say ,
45x^2 +15x +1
45x^2 +15x - 1 ( 5x^2+5x +/-1 where x=3*n )
And how does that help?SlowThinker said:For the original question, the first step is to write the equation modulo 10.
Janosh89 said:This is applicable to a sub-class of prime pairs, 59;61 say ,
45x^2 +15x +1
45x^2 +15x - 1 ( 5x^2+5x +/-1 where x=3*n )
These have the same factors , being derived from 20x^2-1 !