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For $p\equiv 1(\mathrm{mod}4)$, part (i) constructs a solution $a=q!$ to $a^2\equiv -1$. It is nonzero modulo $p$, so $a$ and $-a$ are distinct for odd $p$. Any other solution $x$ obeysThe Euclid lemma forces $x\equiv a$ or $-a$. Hence there are exactly two solutions. For $p\equiv 3(\mathrm{mod}4)$, a solution to $x^2\equiv -1$ would have $x^4\equiv 1$, but part (ii) would imply $x^2\equiv 1$, impossible because $p$ cannot divide two. ThereforeThese two cases exhaust all odd primes.

The modular congruence $x^4\equiv 1$ first implies $x\not\equiv 0$. Multiplication by such an $x$ permutes the nonzero residue classes modulo the prime $p$: if $xa\equiv xb$, cancellation gives $a\equiv b$. Multiply all these residues and cancel their nonzero product to obtain $x^{p-1}\equiv 1$, the usual short proof of Fermat little theorem.

Now $(x^2-1)(x^2+1)\equiv 0$. By the Euclid lemma, either $x^2\equiv 1$ or $x^2\equiv -1$. In the latter case, because $p-1=4k+2$,contradicting the preceding product argument, since $p$ is odd. ThusNo unproved assertion about the number of fourth roots is needed.

For a prime number $p$, every nonzero residue class has a unique multiplicative inverse modulo $p$: Bezout identity supplies one because its representative is coprime to $p$, and cancellation modulo a prime gives uniqueness. Pair each residue with its inverse. The only unpaired residues satisfy $a^2\equiv 1$, so $p\mid (a-1)(a+1)$; the Euclid lemma gives $a\equiv 1$ or $-1$. For odd $p$, these are distinct. Every other pair contributes one to the product of all nonzero residues, leavingFor $p=2$, $1!\equiv -1(\mathrm{mod}2)$ as well. This proves Wilson theorem in all cases.

For an odd prime, set $q=(p-1)/2$. Pair the upper half of the factorial with the negatives of the lower half:If $p\equiv 1(\mathrm{mod}4)$, then $q$ is even. ThereforeThis constructs a square root of minus one modulo a prime explicitly.

This is not an equivalence relation. With the usual convention that zero is a perfect square, $x^2-x^2=0$ gives reflexivity, and changing the sign of $x^2-y^2$ shows that the relation is symmetric. However $3\sim 5$ because $3^2-5^2=-16=-4^2$, and $5\sim 13$ because $5^2-13^2=-144=-12^2$, whereas $3^2-13^2=-160$ is neither a positive nor a negative perfect square: $12^2<160<13^2$. Thus transitivity fails. If “perfect square” were restricted to positive squares, reflexivity would fail as well, leaving the same negative classification.

This is not an equivalence relation. For $x=1/2$, $x\overline{x}=1/4\notin \mathbb{Z}$, so reflexivity fails. It is symmetric, because $y\overline{x}$ is the conjugate of $x\overline{y}$, and an integer is real. It also fails transitivity: take $x=z=1/2$ and $y=2$. Then $x\overline{y}=y\overline{z}=1$ are integers but $x\overline{z}=1/4$ is not. A single failure would suffice, but these examples identify exactly which properties break.

For nonzero complex numbers, $x\overline{y}=(x/y)|y{|}^2$. Since $|y{|}^2$ is positive real, the condition is equivalent to $x/y\in \mathbb{R}\setminus \{0\}$. This is an equivalence relation. The ratio $x/x=1$ gives reflexivity; the reciprocal of a nonzero real ratio shows that the relation is symmetric; and $(x/y)(y/z)=x/z$ gives transitivity. An equivalence class is a real line through zero with zero removed, or equivalently all nonzero complex numbers with the same argument modulo $\pi$.

This is an equivalence relation. The difference $x-x=0$ is an even integer, proving reflexivity. If $x-y=2k$ then $y-x=-2k$, proving that the relation is symmetric. If $x-y=2k$ and $y-z=2\ell$ then $x-z=2(k+\ell )$, proving transitivity. Its equivalence classes are the translates $x+2\mathbb{Z}$; two real numbers need not be integers themselves to belong to the same class.

An integer square is divisible by four exactly when its base is even: an even base has square $4j^2$, while an odd base has square $(2j+1{)}^2\equiv 1(\mathrm{mod}4)$. Apply this observation to the triangular number $T_n$ obtained above. Since $n(n+1)$ is even,Checking the four residue classes of $n$ modulo four, the products $n(n+1)$ have residues $0,2,2,0$, respectively. ThereforeThe original PDF has two alternatives modulo four; the TeX transcription incorrectly turns the second alternative into a modulus-three expression.

Let $T_n=n(n+1)/2$, the $n$th triangular number. Pairing the terms of the arithmetic progression proves ${\sum }_{r=1}^{n}r=T_n$. We prove the cube identity by mathematical induction. For $n=1$ both sides equal one. If ${\sum }_{r=1}^n{r}^3=T_n^2$, thenAdding $(n+1{)}^3$ establishes the next case. ThusEquivalently, the same displayed difference telescopes from $T_0=0$, providing a direct proof without induction.

The characteristic polynomial has only the eigenvalue $\mu$. If $A$ is diagonalizable, there is an invertible matrix $P$ and a diagonal matrix $D$ with $A=PD{P}^{-1}$. Every diagonal entry of $D$ is an eigenvalue, hence equals $\mu$. Thus $D=\mu I$ and $A=P(\mu I)P^{-1}=\mu I$. Conversely, $\mu I$ is already diagonal in every basis. HenceThis argument uses the existence of an eigenvector basis rather than merely the repeated root of the polynomial; repeated eigenvalues alone do not imply diagonalizability.

Using the monic convention for the characteristic polynomial, expand the determinant along the last column:For the eigenvalue $1$, the equations $(A-I)v=0$ give $v_2=v_1$ and $v_3=0$. Thus every corresponding nonzero eigenvector is a nonzero multiple of $(1,1,0{)}^T$. For the eigenvalue $2$, the first two equations give $v_2=2v_1$ and $v_1=0$, so every corresponding nonzero eigenvector is a nonzero multiple of $(0,0,1{)}^T$. ThereforeThese eigenspaces together have dimension $2$, so they cannot provide an eigenvector basis of ${\mathbb{R}}^3$. The matrix is not diagonalizable; the eigenvalue $1$ has algebraic multiplicity $2$ but geometric multiplicity $1$.

Each element of $H$ permutes the three-point set, giving a group homomorphism $\rho :H\to S_3$. To prove injectivity, let an element fix all three points. Conjugate it by the normalizing Möbius transformation from part (i). The conjugate fixes $0,1,\infty$. A Möbius transformation $z\mapsto (az+b)/(cz+d)$ fixing $0$ has $b=0$, and fixing infinity has $c=0$. It is then $z\mapsto kz$; fixing $1$ forces $k=1$. Thus the conjugate, and hence the original transformation, is the identity. The kernel of a group homomorphism is trivial.

For surjectivity, let $(x_1,x_2,x_3)=(\alpha ,\beta ,\gamma )$ and choose any permutation $\sigma$. Let $g$ normalize the ordered triple $(x_1,x_2,x_3)$ and let $g_{\sigma }$ normalize $(x_{\sigma (1)},x_{\sigma (2)},x_{\sigma (3)})$. Then $g_{\sigma }^{-1}g$ sends $x_j$ to $x_{\sigma (j)}$, lies in $H$, and induces the chosen permutation. Thus $\rho$ is onto andFor the normalized set, the six Möbius transformations permuting three points are $z$, $1-z$, $1/z$, $1/(1-z)$, $z/(z-1)$, and $(z-1)/z$. Conjugating by $g$ gives the corresponding six maps for the original set.

If all three points are finite, define the Möbius transformationIts numerator vanishes at $\alpha$, its pole is at $\gamma$, and substitution of $\beta$ gives $1$. The three points are distinct, so the numerator and denominator are not proportional and the associated $2\times 2$ matrix has nonzero determinant.

If one point is infinite, the limiting formulas areIn each case the finite values and the limit at infinity give $g(\alpha )=0$, $g(\beta )=1$, $g(\gamma )=\infty$, and the corresponding determinant is nonzero. Distinctness permits at most one of the points to be infinite. Hence every ordered triple of distinct points on the Riemann sphere can be normalized to $(0,1,\infty )$.

Put $v=x-a$ and decompose it into axial and perpendicular parts, $v=(v\cdot n)n+v_{\perp }$. For the nondegenerate semi-angle $0<\alpha <\pi /2$, the equation is equivalent toThus each plane perpendicular to $n$ at axial distance $s$ from $a$ meets the surface in a circle of radius $|s|\tan \alpha$. Both signs of $s$ are allowed, and $s=0$ gives the vertex. This proves that it is a double circular cone with the specified vertex and axis; $\alpha$ is the angle each generator makes with either direction of the axis.

For the intersection calculation, define the symmetric matrix $Q={\cos }^2\alpha I-n{n}^T$. The two equations are $(x-a_i{)}^{T}Q(x-a_i)=0$. Their difference givesWith $b=a_1-a_2$ and $m=(a_1+a_2)/2$, symmetry of $Q$ makes the last two terms $2m\cdot Qb$. Hence the intersection plane of parallel congruent double cones isThe unnormalized normal vector is $v=Qb=b{\cos }^2\alpha -n(n\cdot b)$. Expanding its squared norm givesEquivalently, if $b=b_{\perp }+(n\cdot b)n$, this is ${\cos }^4\alpha |b_{\perp }{|}^2+{\sin }^4\alpha (n\cdot b{)}^2>0$. Thus normalization is legitimate and yieldsThe assumption of a genuine cone matters: at the degenerate limits $\alpha =0$ or $\pi /2$, the surface collapses respectively to an axis or a plane and this normal need not exist.

Move the origin to the centroid and write the four vertex vectors as $r_1,r_2,r_3,r_4$. Their sum is zero. The midpoints of the first opposite-edge pair have vectors $(r_1+r_2)/2$ and $(r_3+r_4)/2=-(r_1+r_2)/2$, so their distances from the centroid are equal. The same argument applies to the other two opposite-edge pairs.

Choose the three midpoint distances as $u=|r_1+r_2|/2$, $v=|r_1+r_3|/2$, $w=|r_1+r_4|/2$. Expanding the dot products,Identifying these four norms with the vertex distances proves the tetrahedron centroid-to-midpoint sum of squares:The proof uses no orthogonality or regularity assumption on the tetrahedron.

Take $E$ to be continuously differentiable everywhere on ${\mathbb{R}}^3$, as in the question. Its zero curl says ${\partial }_i{E}_j={\partial }_j{E}_i$. Construct a potential of a conservative vector field explicitly byAll segments lie in ${\mathbb{R}}^3$. Differentiating under the integral and using the symmetry of the derivatives,Hence $E=-\nabla \varphi$. This radial integral potential for a curl-free field also works on a star-shaped domain after translating its centre. On an arbitrary domain, zero curl only guarantees local potentials; the everywhere-on-${\mathbb{R}}^3$ interpretation is important for this global assertion.

For the given field, put $q=e^{-x^{2}z}$. The three pairs of derivatives entering its curl areThus the field is irrotational. To find its potential using $E_y=-{\partial }_y\varphi$, integrate ${\partial }_y\varphi =2yq$ to get $\varphi =y^{2}q+\psi (x,z)$. Comparison with $E_x=-{\partial }_x\varphi$ gives ${\psi }_x=0$, and comparison with $E_z=-{\partial }_z\varphi$ gives ${\psi }_z=0$. ThereforeDifferentiating this expression reproduces all three components with the required minus sign; the only freedom on the connected domain is the additive constant.

Use suffix notation, summing repeated indices. The contraction of two Levi-Civita symbols givesAdding the directional derivative $(F\cdot \nabla )G_i=F_j{\partial }_j{G}_i$ cancels the second term. Interchanging $F,G$ gives another pair whose sum is $G_j{\partial }_i{F}_j$. The total right-hand side is thereforethe $i$th component of the gradient of the dot product. Since this holds for each component, the dot-product gradient identity is proved:It holds for any continuously differentiable vector fields; no irrotationality assumption is used.

The second component is resonant. Differentiating $t{e}^{{\lambda }_{2}t}$ supplies the term missing from the exponential ansatz:Thus a particular integral isThe linearly growing oscillatory envelope is forced resonance in a differential equation; it does not contradict bounded oscillation of the complementary function. The worked initial-value solution in the question's root Solution uses this same resonant mechanism.

Try a particular integral along each forcing eigenvector. Since $A{\mathbf{a}}_j={\lambda }_j{\mathbf{a}}_j$,With both denominators nonzero, add the two responses:Only the forcing component's own eigenvalue enters its denominator; a frequency matching a different eigenmode does not create resonance in this already diagonal forcing component.

After multiplication by ${\sec }^{2}x$, recognize both sides as derivatives:The nontrivial first integral is thereforeIt necessarily depends on the derivative, correcting the printed notation $h(x,y)$. A second integration giveson the specified interval. Since $y(0)=0$, $C_2=0$. A finite derivative at zero also implies $y{y}^{\prime }\to 0$ there, forcing $C_1=0$. A positive particular solution is consequentlyThe negative of this function is the other sign choice. Near zero, $-2\log (\cos x)=x^2+x^4/6+O(x^6)$, so the positive branch has $y=x+x^3/12+O(x^5)$ and $y^{\prime }(0)=1$; the negative branch has derivative $-1$.

For the positive branch, $y^{\prime }=\tan x/y>0$. Its second derivative has numerator $-2\log \cos x-{\sin }^{2}x$, multiplied by a positive factor. This numerator is zero at zero and has derivative $2{\sin }^{3}x/\cos x>0$, so the curve is convex for positive $x$. It starts tangent to $y=x$ and rises without bound as $x\uparrow \pi /2$, with $y\sim \sqrt{-2\log (\pi /2-x)}$. The third panel of the sketch above shows this branch.