This explanation would only be good for those that already understand the subject. One struggles to work through the hidden assumptions: the explanation is there with a few pauses and external research. It is not an introduction; thus, the title is click-bait.
Do 1 and 4 count as one (pair), or as 2 separate solutions? I.e., do n and p-n count as one solution, or as two? I.e., do you only need to find the first residue to find the other, or do you need to find the first two residues to find them all?
The question of whether C is a quadratic residue doesn't depend on the actual solution to the congruence x^2 ≡ C. If at least one solution exists, then C is a quadratic residue.
Here is a simpler proof of this fact which viewers might like. :) We know x² ≡ (p - x)² ≡ a (mod p). Let y = p - x. Then p | x² - y² = (x - y)(x + y). Now, p is a prime so p | x - y or p | x + y (Euclid's lemma) and therefore x = y or x + y = p because x, y ∈ {1, 2, ..., p - 1} which proves the claim. Glad to help!
Perfectly pitched, clear, concise and no stumbles. Textbook delivery.
Wow, another really clear explanation from Mu Prime Math
woooow, so easy to understand, thank you!!
Kung ganito kapogi teacher mo kahit complex pa yan papasok talaga ako😁☺
Best introduction out there
Thanks! This helped a lot.
that was really nice explanation. I got the right lecture. love
Thanks!
thank you so much
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Wonderfull my friend
This explanation would only be good for those that already understand the subject. One struggles to work through the hidden assumptions: the explanation is there with a few pauses and external research. It is not an introduction; thus, the title is click-bait.
More example pls
Small suggestion: you have not proved that quadratic residues do not appear more than 2 times
That's correct! You can prove that there are at most 2 solutions using Lagrange's theorem for polynomial congruences.
Do 1 and 4 count as one (pair), or as 2 separate solutions? I.e., do n and p-n count as one solution, or as two? I.e., do you only need to find the first residue to find the other, or do you need to find the first two residues to find them all?
The question of whether C is a quadratic residue doesn't depend on the actual solution to the congruence x^2 ≡ C. If at least one solution exists, then C is a quadratic residue.
Here is a simpler proof of this fact which viewers might like. :)
We know x² ≡ (p - x)² ≡ a (mod p). Let y = p - x. Then p | x² - y² = (x - y)(x + y).
Now, p is a prime so p | x - y or p | x + y (Euclid's lemma) and therefore x = y or x + y = p because x, y ∈ {1, 2, ..., p - 1} which proves the claim.
Glad to help!