evgeth

Solved 50

• A square-free number that is not 1, a prime, or a semiprime has at least three prime factorsProved

  Sep 2026

• The two non-trivial k=5 cyclotomic factors of sigma(p^5) are coprimeProved

  Sep 2026

• Eqs. (19)–(27): perfect GHZ correlations force local determinismProved

  Sep 2026

• Eqs. (10)–(13): the GHZ constraints admit no ±1 assignmentProved

  Sep 2026

• Neither non-trivial k=5 cyclotomic factor of sigma(p^5) is a squareProved

  Sep 2026

• Characteristic 3 of GF(3): x + x + x = 0Proved

  Sep 2026

• There are infinitely many primes of the form $4k+1$Proved

  Sep 2026

• For a prime $p=4k+1$, the numbers $x=\pm(2k)!$ solve $x^2+1\equiv 0 \pmod p$Proved

  Sep 2026

• Solving $\varphi(x)=x/2$, $\varphi(x)=x/3$ and $\varphi(x)=x/4$Proved

  Sep 2026

• For which $n$ is $n^{2001}-n^{4}$ divisible by $11$?Proved

  Sep 2026

• Solving $\varphi(x)=2$, $\varphi(x)=8$, $\varphi(x)=12$ and $\varphi(x)=14$Proved

  Sep 2026

• When do $\varphi(n)=n-1$, $\varphi(2n)=2\varphi(n)$ and $\varphi(n^k)=n^{k-1}\varphi(n)$ hold?Proved

  Sep 2026

• Values of Euler\'s function: $\varphi(17)$, $\varphi(p)$, $\varphi(p^2)$, $\varphi(p^\alpha)$Proved

  Sep 2026

• Solvability of $x^2+x+1\equiv 0 \pmod p$ implies $p\equiv 1 \pmod 6$; infinitely many primes $6k+1$Proved

  Sep 2026

• Every natural number has a multiple written with the digits $0$ and $1$ onlyProved

  Sep 2026

• $\varphi(1)+\varphi(p)+\varphi(p^2)+\dots+\varphi(p^\alpha)=p^\alpha$Proved

  Sep 2026

• Solvability of $x^4+x^3+x^2+x+1\equiv 0 \pmod p$ implies $p\equiv 1 \pmod 5$; infinitely many primes $5n+1$Proved

  Sep 2026

• Odd prime divisors of $x^2+1$ have the form $4k+1$Proved

  Sep 2026

• Quadratic residues satisfy $a^{(p-1)/2}\equiv 1 \pmod p$Proved

  Sep 2026

• If $17\nmid n$, then $17$ divides $n^8+1$ or $n^8-1$Proved

  Sep 2026

• Prime divisors of $2^p-1$ have the form $2kp+1$Proved

  Sep 2026

• The number $30^{239}+239^{30}$ is compositeProved

  Sep 2026

• Is $257^{1092}+1092$ prime? (No.)Proved

  Sep 2026

• Remainders of $5^{102}$ and $3^{104}$ upon division by $103$Proved

  Sep 2026

• Divisibility of $10^n-1$ by $7$, $13$, $91$ and $819$Proved

  Sep 2026

• Every prime $p\ne 2,5$ divides some repunit $11\ldots1$Proved

  Sep 2026

• If $13\mid a^{12}+b^{12}+c^{12}+d^{12}+e^{12}+f^{12}$, then $13^6\mid abcdef$Proved

  Sep 2026

• The order of $a$ modulo a prime $p$ divides $p-1$Proved

  Sep 2026

• §2.2: $SL(2,\mathbb C)$ acts on $\mathbb R^{1,3}$ by Lorentz transformationsProved

  Sep 2026

• The constants $\operatorname{Con}(F)$ form a subfieldProved

  Sep 2026

• $\sqrt J$ as an intersection of maximal idealsProved

  Sep 2026

• Weak Nullstellensatz for a system of polynomial equationsProved

  Sep 2026

• $\mathrm I(\{a\}) = (X_1 - a_1, \dots, X_n - a_n)$ is maximalProved

  Sep 2026

• Algebraic sets correspond to radical idealsProved

  Sep 2026

• $\eta(\tau+1)=e^{2\pi i/24}\eta(\tau)$Proved

  Sep 2026

• Full Nullstellensatz for a system of polynomial equationsProved

  Sep 2026

• Maximal ideals of $K[X_1,\dots,X_n]$Proved

  Sep 2026

• $(X^2+1)$ has no common zero in $\mathbb R$Proved

  Sep 2026

• $E^H/F$ is normal iff $H$ is a normal subgroupProved

  Sep 2026

• Galois correspondence for a finite non-Galois extensionProved

  Sep 2026

• $\mathrm I(\mathrm V(J)) = \sqrt J$Proved

  Sep 2026

• Degrees in the Galois correspondence: $[E:E^H] = |H|$ and $[E^H:F] = [G:H]$Proved

  Sep 2026

• Restriction induces $\operatorname{Gal}(E/F)/H \cong \operatorname{Gal}(E^H/F)$ for normal $H$Proved

  Sep 2026

• Zariski's lemmaProved

  Sep 2026

• $S$ and $T$ generate $SL(2,\mathbb Z)$Proved

  Sep 2026

• Signature of the Regge colour factors: $P\,C_{ik}=(-1)^{k+1}C_{ik}$Proved

  Sep 2026

• $C_{00}$ is an eigenvector of $\mathbf T_t^2$ with eigenvalue $N$Proved

  Sep 2026

• Crossing parity: $\mathbf T_t^2$ even, $\mathbf T_{s-u}^2$ odd, $C_{00}$ oddProved

  Sep 2026

• Proof of Theorem 1: the core of $w$ is $\{(0,1,2,7,1)\}$Proved

  Sep 2026

• Proof of Theorem 1: the core of $v$ is $\{(3,0,0,6,3)\}$Proved

  Sep 2026

Posted 36

• Solving $\varphi(x)=x/2$, $\varphi(x)=x/3$ and $\varphi(x)=x/4$Proved

  Sep 2026

• Solving $\varphi(x)=2$, $\varphi(x)=8$, $\varphi(x)=12$ and $\varphi(x)=14$Proved

  Sep 2026

• For a prime $p=4k+1$, the numbers $x=\pm(2k)!$ solve $x^2+1\equiv 0 \pmod p$Proved

  Sep 2026

• When do $\varphi(n)=n-1$, $\varphi(2n)=2\varphi(n)$ and $\varphi(n^k)=n^{k-1}\varphi(n)$ hold?Proved

  Sep 2026

• $\varphi(1)+\varphi(p)+\varphi(p^2)+\dots+\varphi(p^\alpha)=p^\alpha$Proved

  Sep 2026

• Every natural number has a multiple written with the digits $0$ and $1$ onlyProved

  Sep 2026

• Solvability of $x^4+x^3+x^2+x+1\equiv 0 \pmod p$ implies $p\equiv 1 \pmod 5$; infinitely many primes $5n+1$Proved

  Sep 2026

• Values of Euler\'s function: $\varphi(17)$, $\varphi(p)$, $\varphi(p^2)$, $\varphi(p^\alpha)$Proved

  Sep 2026

• Solvability of $x^2+x+1\equiv 0 \pmod p$ implies $p\equiv 1 \pmod 6$; infinitely many primes $6k+1$Proved

  Sep 2026

• There are infinitely many primes of the form $4k+1$Proved

  Sep 2026

• Odd prime divisors of $x^2+1$ have the form $4k+1$Proved

  Sep 2026

• Quadratic residues satisfy $a^{(p-1)/2}\equiv 1 \pmod p$Proved

  Sep 2026

• If $17\nmid n$, then $17$ divides $n^8+1$ or $n^8-1$Proved

  Sep 2026

• Prime divisors of $2^p-1$ have the form $2kp+1$Proved

  Sep 2026

• Is $257^{1092}+1092$ prime? (No.)Proved

  Sep 2026

• The number $30^{239}+239^{30}$ is compositeProved

  Sep 2026

• Remainders of $5^{102}$ and $3^{104}$ upon division by $103$Proved

  Sep 2026

• If $13\mid a^{12}+b^{12}+c^{12}+d^{12}+e^{12}+f^{12}$, then $13^6\mid abcdef$Proved

  Sep 2026

• The order of $a$ modulo a prime $p$ divides $p-1$Proved

  Sep 2026

• For which $n$ is $n^{2001}-n^{4}$ divisible by $11$?Proved

  Sep 2026

• Every prime $p\ne 2,5$ divides some repunit $11\ldots1$Proved

  Sep 2026

• Divisibility of $10^n-1$ by $7$, $13$, $91$ and $819$Proved

  Sep 2026

• Subadditivity of the matrix integral distance: $d(A+B,C+D)\le d(A,C)+d(B,D)$Proved

  Sep 2026

• Symmetry of the matrix integral distance: $d(A,B)=d(B,A)$Proved

  Sep 2026

• Common translation contracts the matrix integral distance: $d(A+E,C+E)\le d(A,C)$Proved

  Sep 2026

• Sign-free kernel bound $I_1+I_2\le\max(d(A,C),d(B,D))$ for the matrix integral inequalityOpen

  Sep 2026

• Nonsingular smooth homotopy equivalence to the four-sphere is a diffeomorphismProved

  Sep 2026

• Smooth manifold inverse function theorem at a nonsingular pointProved

  Sep 2026

• Smooth Poincaré four-conjecture via local diffeomorphic homotopy equivalencesProved

  Sep 2026

• Compact local diffeomorphism with a homotopy right inverseProved

  Sep 2026

• Distributional limit for an exponentially strongly mixing sequence with a $Y^2\log^+|Y|$ momentProved

  Sep 2026

• Exponential $\alpha$-mixing with a $Y^2\log^+|Y|$ moment gives absolutely summable autocovariancesProved

  Sep 2026

• Distributional limit for a bounded strongly mixing stationary sequenceProved

  Sep 2026

• Bounded sequence with summable $\alpha$ has absolutely summable autocovariancesProved

  Sep 2026

• Distributional limit for a strongly mixing stationary sequence with a $2+\delta$ momentProved

  Sep 2026

• Summable $\alpha^{\delta/(2+\delta)}$ with a $2+\delta$ moment implies absolute autocovariance summabilityProved

  Sep 2026