Number Theory | Quadratic Residues: Definition and Examples

1. Formula
   Observation

   The board writes `$a \in \mathbb{Z}$` in the definition.

2. Audio
   Observation

   The speaker says, "if a and n are integers".

a

An integer whose quadratic-residue status modulo n is being discussed.

$\mathbb{Z}$

1. Formula
   Observation

   The board writes `$n > 0$` and later uses `$\pmod n$` and `$\pmod 7$`.

2. Audio
   Observation

   The speaker says, "n is in addition bigger than zero, so it's a natural number".

n

Positive modulus; in the example it is specialized to 7.

$\mathbb{Z}$,\ $n>0$

1. Formula
   Observation

   The board writes `$\gcd(a,n)=1$`.

2. Audio
   Observation

   The speaker says, "the gcd of a and n is one, so they're coprime".

$\gcd(a,n)$

Greatest common divisor of a and n; the definition requires it to equal 1.

Integer-valued function on pairs of integers

1. Formula
   Observation

   The board writes `$x^2 \equiv a \pmod n$`.

2. Audio
   Observation

   The speaker says, "if x squared congruent to a mod n is solvable".

x

Unknown integer whose square is tested for congruence to a modulo n.

Integers (implicit from the congruence equation)

1. Formula
   Observation

   The board repeatedly uses `$\equiv$ ... $\pmod n$` and `$\pmod 7$`.

2. Audio
   Observation

   The speaker reads examples such as "congruent to 1 mod 7".

$\equiv \pmod n$

Congruence relation modulo n.

Defined on integers

1. Formula
   Observation

   Left board definition reads: "Let a ∈ ℤ with $n > 0$ and gcd(a,n)=1. Then a is said to be a quadratic residue modulo n if x² ≡ a (mod n) is solvable."

a

An integer in the definition of quadratic residue/non-residue; in the worked examples the specific values are the bases being squared or the residues being classified.

Integers, with the examples using positive residues less than the modulus.

1. Formula
   Observation

   Definition line includes "$n > 0$" and the congruence is written modulo n.

n

The modulus in the general definition of quadratic residue modulo n.

Positive integers.

1. Formula
   Observation

   Definition line includes "gcd(a,n)=1".

gcd(a,n)

Greatest common divisor of a and n; the definition requires it to equal 1.

Integer-valued function on pairs of integers.

1. Formula
   Observation

   Definition line includes "x² ≡ a (mod n) is solvable".

$x^2 \equiv a \pmod n$

Congruence condition whose solvability determines whether a is a quadratic residue modulo n.

Congruence relation over integers modulo n.

1. Formula
   Observation

   Middle column heading reads "ex: $n = 7$".

$n=7$

Modulus used in the first worked example.

Specific positive integer modulus.

1. Formula
   Observation

   After erasing the first example, the middle column heading is rewritten as "ex: $n = 8$".

$n=8$

Modulus used in the second worked example.

Specific positive integer modulus.

1. Formula
   Observation

   Right column conclusion reads "So 1,2,4 are quadratic residues mod 7."

1,2,4

Quadratic residues modulo 7 identified from the listed squares.

Residue classes modulo 7.

1. Formula
   Observation

   Board text: "Definition: Let $a \in \mathbb{Z}$ with $n>0$ and $\gcd(a,n)=1$. Then a is said to be a quadratic residue modulo n if $x^2 \equiv a \pmod n$ is solvable. Otherwise it is a quadratic non-residue."

2. Audio
   Observation

   Speaker states the same definition aloud, including the correction from "a squared" to "x squared".

For an integer a and a positive modulus n with gcd(a,n)=1, a is called a quadratic residue modulo n exactly when the congruence $x^2 \equiv a \pmod n$ has a solution. If that congruence has no solution, a is called a quadratic non-residue modulo n.

1. a is an integer

2. $n>0$

3. $\gcd(a,n)=1$

4. solvability means there exists an integer x satisfying the congruence

1. Audio
   Observation

   Speaker says, "we'll mostly be interested when n is an odd prime."

Although the definition is stated for any positive modulus n coprime to a, the lecturer announces that the later theory will mainly concentrate on the case where n is an odd prime.

1. General definition allows any $n>0$ with $\gcd(a,n)=1$

2. Subsequent emphasis is on odd prime n

1. Quadratic residue and quadratic non-residue modulo n

1. Formula
   Observation

   Left board states: "Definition: Let a ∈ ℤ with $n > 0$ and gcd(a,n)=1. Then a is said to be a quadratic residue modulo n if x² ≡ a (mod n) is solvable."

2. Audio
   Observation

   The speaker continues the definition by saying that otherwise it is a quadratic non-residue.

The video gives a definition for an integer a relative to a positive modulus n, under the explicit coprimality condition gcd(a,n)=1. It says a is a quadratic residue modulo n exactly when the congruence x² ≡ a (mod n) has a solution. The same setup also defines the complementary case: if that congruence is not solvable, then a is called a quadratic non-residue modulo n.

1. a ∈ ℤ

2. $n > 0$

3. gcd(a,n)=1

4. Classification depends on whether x² ≡ a (mod n) is solvable

1. Formula
   Observation

   Left board includes: "Otherwise, it is a quadratic non-residue."

2. Audio
   Observation

   In both examples the speaker labels the leftover coprime residues as quadratic non-residues.

Within the same definition block, the video states the negative case: if a satisfies gcd(a,n)=1 but the congruence x² ≡ a (mod n) is not solvable, then a is a quadratic non-residue modulo n. The examples apply this by taking the coprime residues not appearing among the computed square values.

1. a ∈ ℤ

2. $n > 0$

3. gcd(a,n)=1

4. x² ≡ a (mod n) is not solvable

1. Definition of quadratic residue modulo n

1. Audio
   Observation

   Speaker says: "We're going to use the same strategy. So we're going to take all of the numbers that are relatively prime to 8 and square them and see what we get."

2. Formula
   Observation

   For $n=8$ he lists 1,3,5,7 and computes their squares modulo 8.

The worked examples use a direct computational method. First list the positive residues less than n that are relatively prime to n. Then square each one and reduce modulo n. The distinct results are the quadratic residues; the remaining coprime residues are the quadratic non-residues. This method is shown explicitly for $n=7$ and again for $n=8$.

1. Use the definition with gcd(a,n)=1

2. Compute x² mod n for representatives x coprime to n

3. Collect distinct square values as residues

1. Definition of quadratic residue modulo n
2. Definition of quadratic non-residue modulo n

1. Formula
   Observation

   Middle column lists 1²≡1, 2²≡4, 3²≡2, 4²≡2, 5²≡4, 6²≡1 (mod 7).

2. Formula
   Observation

   Right column writes: "So 1,2,4 are quadratic residues mod 7."

For modulus 7, the quadratic residues among the coprime residues are 1, 2, and 4.

1. $n=7$

2. Consider residues relatively prime to 7

3. Use the definition of quadratic residue via solvability of x² ≡ a (mod 7)

For the listed residue classes modulo 7.

1. Audio
   Observation

   Speaker says: "whatever's left, so which is 3, 5, and 6, are quadratic non-residues mod 7."

2. Formula
   Observation

   Right column writes: "3,5,6 are quad non-residues mod 7."

For modulus 7, the quadratic non-residues among the coprime residues are 3, 5, and 6.

1. $n=7$

2. Residues 3,5,6 are coprime to 7

3. They do not appear among the square values 1,2,4 modulo 7

For the listed residue classes modulo 7.

1. Formula
   Observation

   Middle column lists 1²≡1, 3²≡1, 5²≡1, 7²≡1 (mod 8).

2. Formula
   Observation

   Right column writes: "1 is the only quadratic residue modulo 8."

For modulus 8, 1 is the only quadratic residue among the coprime residues.

1. $n=8$

2. Consider residues relatively prime to 8

3. Use the definition of quadratic residue via solvability of x² ≡ a (mod 8)

Among the coprime residue classes modulo 8.

1. Audio
   Observation

   Speaker says: "So we have 1 as a quadratic residue, that means 3, 5, and 7 are quadratic non-residues."

For modulus 8, the quadratic non-residues among the coprime residues are 3, 5, and 7.

1. $n=8$

2. Residues 3,5,7 are coprime to 8

3. They are not equal to the sole square value 1 modulo 8

For the listed residue classes modulo 8.

1. Audio
   Observation

   Speaker says, "what we're going to do is just square all of the numbers mod 7 and see what the outputs are."

2. Formula
   Observation

   Board lists `$1^2 \equiv 1$`, `$2^2 \equiv 4$`, `$3^2 = 9 \equiv 2$`, `$4^2 = 16 \equiv 2$`, `$5^2 = 25 \equiv 4$`, `$6^2 = 36 \equiv 1$` all `($\bmod 7$)`.

1. Expression

   $$1^2 \equiv 1 \pmod 7$$
   Explanation

   Square 1 and reduce modulo 7.

   Justification

   Direct computation.

   Shown in the video
2. Expression

   $$2^2 \equiv 4 \pmod 7$$
   Explanation

   Square 2 and reduce modulo 7.

   Justification

   Direct computation.

   Shown in the video
3. Expression

   $$3^2 = 9 \equiv 2 \pmod 7$$
   Explanation

   Square 3 to get 9, then reduce 9 modulo 7.

   Justification

   Direct computation and reduction modulo 7.

   Shown in the video
4. Expression

   $$4^2 = 16 \equiv 2 \pmod 7$$
   Explanation

   Square 4 to get 16, then reduce 16 modulo 7.

   Justification

   Direct computation and reduction modulo 7.

   Shown in the video
5. Expression

   $$5^2 = 25 \equiv 4 \pmod 7$$
   Explanation

   Square 5 to get 25, then reduce 25 modulo 7.

   Justification

   Direct computation and reduction modulo 7.

   Shown in the video
6. Expression

   $$6^2 = 36 \equiv 1 \pmod 7$$
   Explanation

   Square 6 to get 36, then reduce 36 modulo 7.

   Justification

   Direct computation and reduction modulo 7.

   Shown in the video

The residues obtained by squaring 1 through 6 modulo 7 are 1, 4, 2, 2, 4, 1.

1. Audio
   Observation

   Speaker says, "if we fold this list over this line right here they match. So two matches with two, four with four, and one with one."

2. Diagram
   Observation

   The six displayed congruences are arranged vertically, and the speaker gestures toward the middle of the list.

1. Expression

   $$1^2 \leftrightarrow 6^2$$
   Explanation

   Compare the first and last entries of the list.

   Justification

   Both reduce to 1 modulo 7.

   Shown in the video
2. Expression

   $$2^2 \leftrightarrow 5^2$$
   Explanation

   Compare the second and fifth entries.

   Justification

   Both reduce to 4 modulo 7.

   Shown in the video
3. Expression

   $$3^2 \leftrightarrow 4^2$$
   Explanation

   Compare the third and fourth entries.

   Justification

   Both reduce to 2 modulo 7.

   Shown in the video

The list of squares modulo 7 has a mirrored structure: matching pairs give the same residue.

1. Formula
   Observation

   Middle column shows the six square computations modulo 7.

2. Formula
   Observation

   Right column states the resulting residue and non-residue sets.

1. Expression

   $$1^2 \equiv 1 \pmod 7$$
   Explanation

   Square the first coprime representative and reduce modulo 7.

   Justification

   Direct modular arithmetic.

   Shown in the video
2. Expression

   $$2^2 \equiv 4 \pmod 7$$
   Explanation

   Square the next representative and reduce modulo 7.

   Justification

   Direct modular arithmetic.

   Shown in the video
3. Expression

   $$3^2 = 9 \equiv 2 \pmod 7$$
   Explanation

   Compute 9 and reduce it modulo 7.

   Justification

   Reduction of an integer modulo 7.

   Shown in the video
4. Expression

   $$4^2 = 16 \equiv 2 \pmod 7$$
   Explanation

   Compute 16 and reduce it modulo 7.

   Justification

   Reduction of an integer modulo 7.

   Shown in the video
5. Expression

   $$5^2 = 25 \equiv 4 \pmod 7$$
   Explanation

   Compute 25 and reduce it modulo 7.

   Justification

   Reduction of an integer modulo 7.

   Shown in the video
6. Expression

   $$6^2 = 36 \equiv 1 \pmod 7$$
   Explanation

   Compute 36 and reduce it modulo 7.

   Justification

   Reduction of an integer modulo 7.

   Shown in the video
7. Expression

   $$\{1,2,4\}$$
   Explanation

   Collect the distinct square values obtained modulo 7.

   Justification

   By the definition, these are the quadratic residues modulo 7.

   Shown in the video
8. Expression

   $$\{3,5,6\}$$
   Explanation

   Take the remaining coprime residues not appearing in the square-value set.

   Justification

   By the definition, these are the quadratic non-residues modulo 7.

   Shown in the video

Modulo 7, the quadratic residues are 1, 2, 4 and the quadratic non-residues are 3, 5, 6.

1. Formula
   Observation

   Middle column lists 1,3,5,7 and then their squares modulo 8.

2. Audio
   Observation

   Speaker explains which numbers are relatively prime to 8 and then computes the squares.

3. Formula
   Observation

   Right column concludes that 1 is the only quadratic residue modulo 8.

1. Expression

   $$\gcd(1,8)=1,\ \gcd(3,8)=1,\ \gcd(5,8)=1,\ \gcd(7,8)=1$$
   Explanation

   Identify the positive residues less than 8 that are relatively prime to 8.

   Justification

   Direct check of coprimality with 8.

   Shown in the video
2. Expression

   $$1^2 \equiv 1 \pmod 8$$
   Explanation

   Square the first coprime representative.

   Justification

   Direct modular arithmetic.

   Shown in the video
3. Expression

   $$3^2 = 9 \equiv 1 \pmod 8$$
   Explanation

   Compute 9 and reduce modulo 8.

   Justification

   Reduction of an integer modulo 8.

   Shown in the video
4. Expression

   $$5^2 = 25 \equiv 1 \pmod 8$$
   Explanation

   Compute 25 and reduce modulo 8.

   Justification

   Reduction of an integer modulo 8.

   Shown in the video
5. Expression

   $$7^2 = 49 \equiv 1 \pmod 8$$
   Explanation

   Compute 49 and reduce modulo 8.

   Justification

   Reduction of an integer modulo 8.

   Shown in the video
6. Expression

   $$\{1\}$$
   Explanation

   Collect the distinct square values obtained modulo 8.

   Justification

   By the definition, these are the quadratic residues modulo 8.

   Shown in the video
7. Expression

   $$\{3,5,7\}$$
   Explanation

   Take the remaining coprime residues not appearing among the square values.

   Justification

   By the definition, these are the quadratic non-residues modulo 8.

   Shown in the video

Modulo 8, the only quadratic residue among the coprime residues is 1, and the non-residues are 3, 5, 7.

1. Formula
   Observation

   Board writes "ex: $n=7$" and then the six square congruences.

2. Audio
   Observation

   Speaker says, "So let's take n equals 7" and proceeds to square all numbers mod 7.

Determine which residues arise as squares modulo 7 and identify the quadratic residues among them.

1. $n=7$

2. Consider the integers 1,2,3,4,5,6 modulo 7

Compute $x^2 \pmod 7$ for these x-values and read off the quadratic residues.

1. Expression

   $$1^2 \equiv 1 \pmod 7$$
   Explanation

   Compute the square of 1 modulo 7.

   Justification

   Direct calculation.

   Shown in the video
2. Expression

   $$2^2 \equiv 4 \pmod 7$$
   Explanation

   Compute the square of 2 modulo 7.

   Justification

   Direct calculation.

   Shown in the video
3. Expression

   $$3^2 = 9 \equiv 2 \pmod 7$$
   Explanation

   Compute the square of 3 and reduce.

   Justification

   Direct calculation.

   Shown in the video
4. Expression

   $$4^2 = 16 \equiv 2 \pmod 7$$
   Explanation

   Compute the square of 4 and reduce.

   Justification

   Direct calculation.

   Shown in the video
5. Expression

   $$5^2 = 25 \equiv 4 \pmod 7$$
   Explanation

   Compute the square of 5 and reduce.

   Justification

   Direct calculation.

   Shown in the video
6. Expression

   $$6^2 = 36 \equiv 1 \pmod 7$$
   Explanation

   Compute the square of 6 and reduce.

   Justification

   Direct calculation.

   Shown in the video
7. Expression

   $$\{1,2,4\}$$
   Explanation

   Collect the distinct outputs from the six computations.

   Justification

   These are exactly the residues represented by some square modulo 7.

   Derived from the video

The quadratic residues modulo 7 are 1, 2, and 4.

Each listed residue appears among the computed squares modulo 7, and no other nonzero residue appears in the list.

1. Formula
   Observation

   Middle column heading reads "ex: $n = 7$" with six square congruences.

2. Formula
   Observation

   Right column states the residue and non-residue conclusions.

Classify the quadratic residues and non-residues modulo 7 using the definition.

1. $n=7$

2. Coprime residues to consider: 1,2,3,4,5,6

3. Computed squares modulo 7 are shown on the board

Determine which residue classes modulo 7 are quadratic residues and which are quadratic non-residues.

1. Expression

   $$1^2 \equiv 1 \pmod 7$$
   Explanation

   Start with the first representative and reduce its square.

   Justification

   Direct computation.

   Shown in the video
2. Expression

   $$2^2 \equiv 4 \pmod 7$$
   Explanation

   Continue through the representatives.

   Justification

   Direct computation.

   Shown in the video
3. Expression

   $$3^2 = 9 \equiv 2 \pmod 7$$
   Explanation

   Reduce 9 modulo 7.

   Justification

   Direct computation.

   Shown in the video
4. Expression

   $$4^2 = 16 \equiv 2 \pmod 7$$
   Explanation

   Reduce 16 modulo 7.

   Justification

   Direct computation.

   Shown in the video
5. Expression

   $$5^2 = 25 \equiv 4 \pmod 7$$
   Explanation

   Reduce 25 modulo 7.

   Justification

   Direct computation.

   Shown in the video
6. Expression

   $$6^2 = 36 \equiv 1 \pmod 7$$
   Explanation

   Reduce 36 modulo 7.

   Justification

   Direct computation.

   Shown in the video
7. Expression

   $$1,2,4$$
   Explanation

   Read off the distinct square values as the residues.

   Justification

   Definition of quadratic residue via solvability of x² ≡ a (mod n).

   Shown in the video
8. Expression

   $$3,5,6$$
   Explanation

   The remaining coprime classes are the non-residues.

   Justification

   Definition of quadratic non-residue as the unsolvable case.

   Shown in the video

Quadratic residues mod 7: 1, 2, 4. Quadratic non-residues mod 7: 3, 5, 6.

The listed square values modulo 7 are exactly 1, 2, and 4, and the remaining coprime classes are 3, 5, and 6.

1. Formula
   Observation

   Middle column heading reads "ex: $n = 8$" and lists 1,3,5,7 with their squares modulo 8.

2. Audio
   Observation

   Speaker says this is a composite-number example and uses the same strategy.

3. Formula
   Observation

   Right column states: "1 is the only quadratic residue modulo 8."

Classify the quadratic residues and non-residues modulo 8 using the same strategy as before.

1. $n=8$

2. Only residues relatively prime to 8 are considered: 1,3,5,7

3. Squares modulo 8 are computed on the board

Determine which coprime residue classes modulo 8 are quadratic residues and which are quadratic non-residues.

1. Expression

   $$1,3,5,7$$
   Explanation

   Select the positive residues less than 8 that are relatively prime to 8.

   Justification

   The definition requires gcd(a,n)=1, and the speaker explicitly excludes 2,4,6.

   Shown in the video
2. Expression

   $$1^2 \equiv 1 \pmod 8$$
   Explanation

   Square the first selected representative.

   Justification

   Direct computation.

   Shown in the video
3. Expression

   $$3^2 = 9 \equiv 1 \pmod 8$$
   Explanation

   Reduce 9 modulo 8.

   Justification

   Direct computation.

   Shown in the video
4. Expression

   $$5^2 = 25 \equiv 1 \pmod 8$$
   Explanation

   Reduce 25 modulo 8.

   Justification

   Direct computation.

   Shown in the video
5. Expression

   $$7^2 = 49 \equiv 1 \pmod 8$$
   Explanation

   Reduce 49 modulo 8.

   Justification

   Direct computation.

   Shown in the video
6. Expression

   $$1$$
   Explanation

   All square computations give the same residue class.

   Justification

   Distinct square values determine the quadratic residues.

   Shown in the video
7. Expression

   $$3,5,7$$
   Explanation

   The other coprime classes are therefore non-residues.

   Justification

   Definition of quadratic non-residue as the unsolvable case.

   Shown in the video

The only quadratic residue modulo 8 is 1; the quadratic non-residues are 3, 5, 7.

Each coprime base 1,3,5,7 squares to 1 modulo 8, so no other coprime class appears as a square value.

1. Diagram
   Observation

   A blackboard is split into a left definition panel and a right example panel.

2. Animation
   Observation

   The lecturer points to the definition, then writes the example heading and the six congruences on the right side.

1. Left panel with the formal definition

2. Right panel headed "ex: $n=7$"

3. Vertical list of six congruences modulo 7

1. Right panel begins blank

2. Then `ex: $n=7$` is written

3. Then the six square congruences are added one by one

4. Finally the speaker gestures at the symmetry among the rows

1. Left-panel definition remains visible throughout

2. Modulus 7 remains fixed during the example

The visual layout separates the general definition from a concrete computational example, making the transition from abstract criterion to explicit residue testing easy to follow.

1. Diagram
   Observation

   The blackboard is divided into three vertical sections: definition on the left, worked computations in the middle, and conclusions on the right.

1. Left definition panel

2. Middle example-computation panel

3. Right conclusion panel

4. Presenter writing with chalk

1. The middle and right panels are first filled with the $n=7$ example.

2. Around 42 seconds the presenter erases the middle and right panels while leaving the left definition intact.

3. The middle and right panels are then refilled with the $n=8$ example.

1. The left definition remains visible throughout the clip.

2. The mathematical task stays the same: classify quadratic residues and non-residues for a chosen modulus.

The visual layout separates theory from computation and conclusion, making the examples read as applications of the fixed definition on the left.

1. Animation
   Observation

   The presenter wipes away the previous example content from the middle and right columns before rewriting a new heading.

1. Chalkboard eraser/hand motion

2. Old $n=7$ computations

3. New blank middle/right space

1. The $n=7$ computations and conclusions disappear.

2. A new heading for $n=8$ is written in the middle column.

1. The definition on the left does not change.

2. The method of squaring coprime representatives is carried over to the new modulus.

This visual reset marks a shift from an odd prime modulus to a composite modulus while keeping the same definitional framework.

1. Audio
   Observation

   Speaker says, "if a squared is congruent sorry if x squared congruent to a mod n is solvable."

One might think the defining equation is $a^2 \equiv a \pmod n$ because the speaker initially misspeaks.

The corrected and board-written condition is $x^2 \equiv a \pmod n$: x is the unknown being solved for, while a is the integer whose residue status is under discussion.

1. Audio
   Observation

   For $n=8$ the speaker explicitly says: "2 is not relatively prime, so we can't use that. 4 is not, so we can't use that. 6 is not," and only keeps 1,3,5,7.

One might try to classify every residue modulo n as a quadratic residue or non-residue without checking gcd(a,n)=1.

The video's definition applies only when gcd(a,n)=1. In the $n=8$ example, 2, 4, and 6 are excluded before squaring because they are not relatively prime to 8.

1. Formula
   Observation

   For $n=8$, four different bases 1,3,5,7 all produce the same square value 1 modulo 8.

2. Audio
   Observation

   The speaker concludes from the repeated outputs that 1 is the only quadratic residue modulo 8.

One might expect each coprime base to produce a different quadratic residue.

The examples show that several different bases can square to the same residue class. What matters is the set of distinct square values modulo n, not the number of bases used to obtain them.

1. Formula
   Observation

   Left panel gives the general definition; right panel applies it with $n=7$.

2. Audio
   Observation

   Speaker moves from the definition directly into "let's look at an example".

The example with $n=7$ applies the general definition by explicitly computing which residues occur as squares modulo 7.

1. Audio
   Observation

   After stating the general definition, the speaker says future results will mostly concern odd prime n.

Odd prime moduli are a special case of the general definition that the lecturer highlights for later theory.

1. Audio
   Observation

   Speaker notes the folded matching pattern before listing the residues.

2. Formula
   Observation

   The repeated residues 1,4,2 appear in symmetric positions in the six-row list.

The observed mirrored structure helps justify why only three distinct residues, 1, 2, and 4, are collected as the quadratic residues modulo 7.

1. Formula
   Observation

   The left definition introduces solvability of x² ≡ a (mod n).

2. Audio
   Observation

   The speaker says for $n=8$: "We're going to use the same strategy" and then squares the coprime representatives.

The computational method is an application of the definition: by squaring candidate bases and reducing modulo n, one directly tests which coprime residues occur as square values.

1. Formula
   Observation

   The definition states the residue case and then says "Otherwise, it is a quadratic non-residue."

2. Formula
   Observation

   Both examples split the coprime classes into square values and leftover classes.

Quadratic non-residue is defined as the complementary case to quadratic residue under the same coprimality hypothesis.

1. Audio
   Observation

   Speaker says: "we've looked at an example where we have an odd prime, and so now I'm going to do one more example where we have a composite number just for good measure."

2. Formula
   Observation

   Both examples use the same three-step pattern: list coprime classes, square them, classify the outcomes.

The two examples contrast an odd prime modulus with a composite modulus while using the same definition-driven procedure.

1. Formula
   Observation

   The $n=7$ square computations lead directly to the stated residue and non-residue sets.

The claim about residues modulo 7 depends on the method of computing distinct square values from coprime representatives.

1. Formula
   Observation

   The $n=8$ square computations lead directly to the statement that 1 is the only quadratic residue modulo 8.

The uniqueness claim for modulo 8 follows from the explicit computation that all coprime bases square to 1.

1. Formula
   Observation

   Definition written on the left side of the board.

2. Audio
   Observation

   Speaker explains the definition verbally.

1. Quadratic residue and quadratic non-residue modulo n

1. Formula
   Observation

   Board includes `$\gcd(a,n)=1$` in the hypothesis.

2. Audio
   Observation

   Speaker says, "so they're coprime".

1. Quadratic residue and quadratic non-residue modulo n

1. Formula
   Observation

   Six explicit congruences modulo 7 are written on the board.

2. Audio
   Observation

   Speaker says he will square all numbers mod 7 and see the outputs.

1. Example with modulus 7
2. Computing the squares modulo 7

1. Audio
   Observation

   Speaker says the list can be folded and the entries match.

2. Diagram
   Observation

   Vertical arrangement of the six congruences makes the pairing visible.

1. Symmetry observed in the list of squares modulo 7
2. Example with modulus 7

1. Audio
   Observation

   Speaker says later results will mostly be about odd prime n.

1. Usual focus on odd prime moduli

1. Formula
   Observation

   Left board definition is visible throughout.

1. Definition of quadratic residue modulo n

1. Formula
   Observation

   Left board includes the "Otherwise" clause.

1. Definition of quadratic non-residue modulo n

1. Audio
   Observation

   Speaker excludes 2,4,6 for $n=8$ because they are not relatively prime to 8.

1. Definition of quadratic residue modulo n
2. Do not include non-coprime residues in this definition-based classification

1. Formula
   Observation

   Right column states the $n=7$ conclusion.

1. Worked example: quadratic residues modulo 7
2. Quadratic residues modulo 7

1. Formula
   Observation

   Right column states the $n=7$ non-residue conclusion.

1. Worked example: quadratic residues modulo 7
2. Quadratic non-residues modulo 7

1. Formula
   Observation

   Right column states that 1 is the only quadratic residue modulo 8.

1. Worked example: quadratic residues modulo 8
2. Unique quadratic residue modulo 8

1. Audio
   Observation

   Speaker says 3, 5, and 7 are quadratic non-residues.

1. Worked example: quadratic residues modulo 8
2. Quadratic non-residues modulo 8