Let's use induction on $k$. For $k=3$ one can show by computation. For $k>3$, we have by induction that $a^{n/8}\equiv 1\pmod{n/2}$. Thus $a^{n/8}=nm/2+1$. This gives $a^{n/4}= 1+ nm + n^2m/4$. Can you finish?Alexmahone said:Let $n=2^k$ where $k\ge 3$. Let $a$ be any odd natural number.
Prove that $a^{n/4}\equiv 1\pmod{n}$.
My attempt:
$\phi(n)=n/2$
So, by Euler's formula, $a^{n/2}\equiv 1\pmod{n}$.
I don't know how to proceed.
This is probably the best way to do it and explains why we have "$n/4$" in the problem. As always, johng is awesome, especially at group theory.johng said:Since you have posted questions on groups, you might be interested in the following link: https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n. Here, it is stated that for $n=2^k$, the structure of $Z_n^\ast$ is the direct product of the cyclic group of order 2 and the cyclic group of order $2^{k-2}$; it also states that 3 has order $2^{n-2}$ You might try and prove these things for your self or look in practically any book on groups.
Congruence for powers of 2 refers to the property of numbers that have the same remainder when divided by a power of 2. In other words, two numbers are congruent if they have the same last digits when written in binary form.
To prove congruence for powers of 2, we can use the principle of mathematical induction. This involves showing that the property holds for the first number in the sequence, then assuming it holds for an arbitrary number and proving that it also holds for the next number in the sequence. By repeating this process, we can prove that the property holds for all numbers in the sequence.
Proving congruence for powers of 2 is important because it allows us to make deductions and predictions about numbers in binary form. It also has applications in computer science and cryptography, where understanding congruence can help with data encryption and security.
Some examples of congruence for powers of 2 include 5 and 13, as both have the same remainder when divided by 4, and 10 and 26, as both have the same remainder when divided by 8. In general, any two numbers that have the same last digits when written in binary form are congruent for powers of 2.
Yes, there are exceptions to the rule of congruence for powers of 2. For example, the numbers 6 and 14 are not congruent for powers of 2, even though they have the same remainder when divided by 4. This is because 6 in binary form is 110, while 14 in binary form is 1110. Therefore, the last digits do not match, and the numbers are not congruent for powers of 2.