Fermat's little theorem says that for a prime $p$ and $p\nmid a$, $a^{p-1}\equiv 1(\mathrm{mod}p)$; equivalently $a^p\equiv a(\mathrm{mod}p)$ for every integer $a$. The Wilson theorem says $(p-1)!\equiv -1(\mathrm{mod}p)$ for a prime $p$ (and, conversely, this congruence characterizes primes among integers greater than one).

For $p=2$, $x=1$ is a square root of minus one modulo a prime. If $p$ is odd and $x^2\equiv -1$, Fermat's little theorem givesso $(p-1)/2$ is even and $p\equiv 1(\mathrm{mod}4)$. Conversely, put $h=(p-1)/2$. Pairing $j$ with $p-j$ in the factorial gives $(p-1)!\equiv (-1{)}^h(h!{)}^2$. For $p\equiv 1(\mathrm{mod}4)$, $h$ is even, so the Wilson theorem yields $(h!{)}^2\equiv -1$. Thusand in the latter case $x=h!$ supplies a solution.

For the multiplicative order assertion, divide $k$ by $d$: $k=qd+r$, $0\le r<d$. Since $x^d\equiv 1$ and $x^k\equiv 1$, it follows that $x^r\equiv 1$. Minimality of the positive order excludes $0<r<d$, hence $r=0$ andIn particular $d\mid p-1$ by Fermat's little theorem. Negative $k$, if included, are handled by the same division using the modular inverse of $x$.

A Fermat number in the paper has $n\ge 1$. If a prime $p$ divides $F_n=2^{2^n}+1$, it is odd and $2^{2^n}\equiv -1(\mathrm{mod}p)$. Squaring gives $2^{2^{n+1}}\equiv 1$, so the order $d$ divides $2^{n+1}$. It does not divide $2^n$, since $-1\ne 1$ modulo an odd prime. Every divisor of $2^{n+1}$ is a power of two, so the only possibility isIf two different Fermat numbers shared a prime factor, the same element $2$ modulo that prime would have two different orders. This is impossible, proving pairwise coprimality. Also $2^{n+1}\mid p-1$, so for $n\ge 1$ every such prime is $1$ modulo $4$. No prime $3$ modulo $4$ can occur.

A prime $1$ modulo $4$ need not occur: take $p=13$. It is prime, $2^6=64\equiv -1(\mathrm{mod}13)$, and $2^4\equiv 3\ne 1$. Since the order divides $12$, the first congruence excludes every divisor of $6$ and the second excludes $4$; hence the order is $12$. This is not a power of two, so $13$ is not a prime divisor of a Fermat number. The index restriction matters: the conventional extra number $F_0=3$ is outside the paper's positive-$n$ family.

For $0<v<c$, the Lorentz transformation with coincident origins isFor two simultaneous events in $S$, $\Delta t=0$, so $\Delta t^{\prime }=-\gamma v\Delta x/c^2$. Distinct spatial positions therefore generally give different times in the boosted frame: this is relativity of simultaneity.

Choose the emission events at $(t,x)=(0,0)$ and $(0,d)$. After emission, the photon world lines are $x=ct$ and $x=d+ct$. Applying the Lorentz transformation to either givesThus the two photons move at speed $c$ in $S^{\prime }$ on parallel lines whose intercepts differ by $\gamma (1+v/c)d$. Measuring their positions at the same $t^{\prime }$ after both emissions gives the photon separation under a collinear Lorentz boost:This separation is constant. It is not the contracted distance between the stationary sources: the two emission events are not simultaneous in $S^{\prime }$, and one photon has already moved when the other is emitted.

By Newton's third law, the force on the second particle is $-F_{12}$. For $M=m_1+m_2$ and centre of mass $R=(m_1{r}_1+m_2{r}_2)/M$, Newton's second law gives $M\ddot{R}=F_{12}-F_{12}=0$. ThusThe centre moves on a straight line with constant velocity; zero velocity is allowed. For the relative position $r=r_1-r_2$, subtraction of the two equations givesHence the two-body problem reduces to a single particle of reduced mass $\mu$, together with the elementary centre motion. Reconstruct the positions as $r_1=R+(m_2/M)r$, $r_2=R-(m_1/M)r$.

Yes: the two equal masses can follow the same fixed circle in diametrically opposite positions. Put the stationary centre of mass at the circle's center and choose tangential velocities with the same sense of rotation. If the circle has radius $a$, the separation is $2a$ and the mutual gravitational force is directed toward that center, with magnitude $G{m}^2/(4a^2)$. The circular orbit condition is $ma{\omega }^2=G{m}^2/(4a^2)$, so an equal-mass circular binary hasThese initial data maintain the opposite positions and give the same constant angular speed for both particles.

For an oriented smooth surface $S$ with piecewise smooth, consistently oriented boundary, Stokes theorem givesfor a continuously differentiable vector field defined on a neighborhood of $S$. Choose the upward normal vector here. The surface is an annular band of an elliptic paraboloid, parametrized byUsing ${\partial }_{r}r\times {\partial }_{\theta }r$ as the oriented vector area element givesThe sketch below shows the open band, not a capped solid. Its upper circle has radius one and its lower circle radius one third.

Differentiating the given vector field yieldsThe surface integral is thereforeFor the boundary line integral, the upward normal vector induces counterclockwise traversal of the outer circle and clockwise traversal of the inner circle, as viewed from above. On a circle of radius $R$, parametrized counterclockwise, $z$ is constant andSince each fourth power integrates to $3\pi /4$, the two-circle line integral isconfirming Stokes theorem and the Stokes flux through an annular paraboloid. Reversing the chosen orientation changes both integrals to $-40\pi /27$.

Work with $x$ modulo $p-1$ and $u$ modulo $p$. By Fermat's little theorem, $a^{p-1}=1$ in ${\mathbb{F}}_p$, so the factor $a^y$ is independent of the chosen representative of $y$. The proposed multiplication is therefore well defined on the stated Cartesian product. Its two associative bracketings have the same second component:and both have first component $x+y+z$. The identity element is $(0,0)$, and the two-sided inverse element isThus the multiplication defines the twisted cyclic pair group, of order $p(p-1)$.

If $a=1$, multiplication is coordinatewise addition, so the group is abelian. If $a\ne 1$, then $p>2$ and both $(1,0)$ and $(0,1)$ are available; their products in opposite orders are $(1,1)$ and $(1,a)$. They differ. HenceBoth $H$ and $K$ contain the identity element and are closed under the multiplication and inverses: they are the cyclic groups of orders $p-1$ and $p$ respectively. Conjugation gives, with all coordinates reduced in the appropriate modulus,so $K$ is a normal subgroup. For $H$, the corresponding calculation isWhen $a=1$ this lies in $H$. Conversely, for $a\ne 1$, take $x=0$, $u=1$, $y=1$: the second coordinate is $a-1\ne 0$, so $H$ is not normal. Thus $H$ is normal precisely when $a=1$.

Finally, the projectionis a surjective group homomorphism because the first coordinates add. Its kernel of a group homomorphism is exactly $K$. At $p=2$, $a=1$ is the only possibility and the first factor is trivial, consistent with every conclusion. No assumption that $a$ generates the multiplicative group of the finite field is needed.