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 !
This is because the quadratic expression has been specifically constructed to have a remainder of 1 when divided by these particular numbers. This is known as a congruence relation, where the remainder of a division is equal to a given number. In this case, the remainder is always 1 when divided by 11, 19, 29, and so on.
Yes, these numbers are known as prime numbers, which means they can only be divided by 1 and themselves. This property makes them useful for constructing congruence relations, as seen in the quadratic expression 20*x^2-1.
Yes, other numbers can be used. However, the numbers must be chosen carefully in order to maintain the congruence relation. Also, the numbers used must be relatively prime to the divisor (11, 19, 29, etc.), meaning they have no common factors other than 1.
Congruence is used in various areas of mathematics, such as number theory and algebra, to study patterns and relationships between numbers. It is also used in cryptography to create secure codes and in geometry to determine if two shapes are identical or have the same properties.
Yes, this concept can be applied in fields such as computer science and engineering, where congruence relations are used in algorithms and coding. It can also be used in data encryption to create secure communication channels.