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In wikipedia source: https://en.wikipedia.org/wiki/Quadratic_residue
under "composite modulus" section
I found the line
"On the other hand, if we want to know if there is a solution for x less than some given limit c, this problem is NP-complete;however, this is a fixed-parameter tractable...
for a given range of x in Zn , and n is composite , and ax² + bx + c ≡ 0(mod n) and if (4a,n)=1,
I learned that we can solve the congruence by (2ax + b)² ≡ b²-4ac (mod n) ==> y² ≡ z (mod n)
So, if n is composite,
Sometimes I see, modulo 4an, when do we take 4an and n ,
how can we prove ...
If given a 'n' value and m = floor ( squareroot(n) )
then is there any way to find the value of 'y' , such that
((m*y) mod n) is congruent to (n-1)
---------- Post added at 07:02 ---------- Previous post was at 05:38 ----------
with the help of a friend
i figured out that, if m is the...
if f(x)modg(x) is valid(means , if it yield a remainder) then , can there be negative powers of x in f(x)?
for example
is (x-29)mod(x2 - 3) possible ?
can we do modulo division like this or is it strictly defined only for positive powers of x?
I know , how to calculate " x mod p " for a given very large "x" value and some p value. Here , x and p are integers
But how can we calculate f(x) mod g(x) , for which f(x) has higher degree(or order) than g(x).
And that degree turns out to be a very large number.
Ex:[ (x+1)^1729 mod ((x^5)-1)...
I did not say for any given (a,b) there surely exits (m,n)
I said " If " p is even and " if " p = a^2 + b^2 = m^2 + n^2 , then what is the relation between a,b,m,n other than p = a^2 + b^2 = m^2 + n^2 equation.
If an Even number could be expressed in the form a² + b² . And if there exits two other numbers m,n such that
a² + b² = m² + n²
then , my question is
is there any relation between (a,b) and (m,n) apart from a² + b² = m² + n² ??
can we find the value of gcd(x c y , z) easily and very fast using a computer.
where
1. "c" represents "combinations" used in 'permutations and combinations'.
2. x is very very large number (ex: may be of 100 or 1000 numerical digits)
3. y is also large having 2 to 5 digits less than x.
4. z...
If y = f(x) , and ( dy / dx ) + ( y / ( sqrt(a+(x^2)) ) ) = 0
I knew its solution is y = { sqrt(a+(x^2)) - x } , where a is a constant
can any one give the proof , by solving the differntial equation.
Are there any other solutions for the above given differential equation. I asked this other...
I like to give an a method for factorization.
Below method is not an efficient method , but it is a method to factorize.
I dont know it already exist or not , which mean I have searched for this process on internet , But I did not find this method. So , may be I missed the page where this...
I have written a C code program for factorization of a given number into two divisors, when the two divisors are having approximately the same number of digits . It cannot check for prime numbers.
With my code limit I can check it only upto 14 digits using long long int. It is working perfectly...
We know If two or more resistances connected in series with an applied voltage 'V', then the equivalent resistance of the circuit is the sum of all the resistances. This is because all the resistors generates the same current.
==> V=IR1+IR2;
But why all the resistors in series are generating...
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