Mistral: on the 100-problem benchmark

Find the inflection points, as ordered pairs, and intervals of concavity of \(f \! \left(x \right) = x^{4}-8 x^{2}+3\).

Analysis: The groups disagree on the x-coordinates of the inflection points (run 1 and run 3 use \(\pm\sqrt{2/3}\), run 2, run 4 and run 5 use \(\pm\frac{2}{3}\sqrt{3}\), run 6 uses \(\pm\frac{2}{3}\sqrt{6}\)), on the y-values (\(-47/9\), \(-59/3\), \(-61/9\), \(-59/9\), \(-163/9\)), on whether \((0,3)\) is reported as an extra inflection point, and on the resulting up/down pattern of the concavity intervals.

Apply 3 iterations of Gauss-Seidel to the system \(10 x_{1}-x_{2}+2 x_{3} = 6\), \(-x_{1}+11 x_{2}-x_{3} = 25\), \(2 x_{1}-x_{2}+10 x_{3} = -11\), starting from \([x_{1}, x_{2}, x_{3}] = [0, 0, 0]\). Give \([x_{1}, x_{2}, x_{3}]\) after iteration 3, each to 6 decimal places.

Analysis: Only runs 1, 4, and 5 match exactly. Runs 2, 3, and 6 each give different iteration-3 vectors, differing in at least one component at the required 6-decimal precision.

Assuming \(0<a\), compute the curvature \(\kappa \! \left(t \right)\) and torsion \(\tau \! \left(t \right)\) of the helix \(r \! \left(t \right) = \langle a \cos \! \left(t \right), a \sin \! \left(t \right), b t \rangle\).

Analysis: The answer from run 1 gives \(\kappa = \frac{a}{a^2+b^2}\), while run 4 gives \(\kappa = \frac{b^2}{a(a^2+b^2)}\), which are different expressions (the torsion \(\tau = \frac{b}{a^2+b^2}\) agrees in both).

Find the closed form of the infinite sum \(\overset{\infty}{\underset{n =1}{\sum}}\frac{H_{n}}{n^{2}}\), where \(H_{n} = \overset{n}{\underset{k =1}{\sum}}\frac{1}{k}\) is the n-th harmonic number. Express the result in terms of the Riemann zeta function.

Analysis: The groups evaluate to \((3/2)\zeta(3)\), \(\zeta(3)+\pi^2\ln 2/6\), \(\zeta(3)+\zeta(2)^2/2\), \(2\zeta(3)\), \((3/2)\zeta(3)+\pi^2\ln 2/2-\pi^2\ln^2 2/2+\ln^4 2/2+12\,\mathrm{LerchPhi}(1,4,1/2)\), and \(\zeta(3)+\pi^2\ln 2/12\), all distinct.

Evaluate \(\int_{0}^{1}\ln \! \left(x \right) \ln \! \left(1-x \right)d x\) in closed form.

Analysis: Run 1 gives \(2 - 2\zeta(2)\) while run 2 gives \(2\zeta(3) - 2\), which are different values since \(\zeta(2) = \pi^2/6 \neq \zeta(3)\).

Find the closed form of the infinite sum \(\overset{\infty}{\underset{n =1}{\sum}}\frac{\cos \! \left(n \right)}{n^{2}}\) (the argument is in radians, not degrees).

Analysis: The six expressions evaluate to six distinct numerical values (approximately \(0.986\), \(-2.758\), \(-2.674\), \(1.035\), \(-0.670\), \(7.83\)), so no two are mathematically equivalent.

Decompose into partial fractions and evaluate \(\int \frac{3 x^{2}+2 x +1}{\left(x -1\right) \left(x +2\right)^{2}}d x\).

Analysis: The four answers have different coefficients on \(\ln(x-1)\), \(\ln(x+2)\), \(1/(x+2)\), and \(1/(x+2)^2\): run 3 gives \(1,\,-5/2,\,-2,\,-11/2\); run 1 gives \(3/2,\,-11/9,\,-11/3,\,0\); run 5 gives \(1/3,\,-7/9,\,-7/3,\,11/9\); run 6 gives \(1/3,\,-11/9,\,-11/3\) plus a non-constant \((-3x-4)/(3(x+2)^2)\) term, none of which differ by an additive constant.

Evaluate \(\int_{0}^{1}\frac{\ln \! \left(x \right)}{\sqrt{-x^{2}+1}}d x\) in closed form.

Analysis: Run 4's answer is \(-\pi^2/8\) while run 2's answer is \(-\frac{1}{2}\pi\ln 2\), which are numerically and symbolically distinct values.

Find the critical points of \(f \! \left(x , y\right) = x^{3}+y^{2}-3 x\) and classify them.

Analysis: Run 5 classifies \((1,0)\) as a local minimum, \((-1,0)\) as a local maximum, and includes \((0,0)\) as a saddle, while run 1 classifies \((1,0)\) as a local minimum, \((-1,0)\) as a saddle point, and omits \((0,0)\).

Locate and classify the critical points of the surface defined by \(f \! \left(x , y\right) = x^{3}+y^{3}-3 x y\).

Analysis: Run 1 and run 6 report two critical points, \((0,0)\) as a saddle and \((1,1)\) as a local minimum, whereas run 5 reports \((0,0)\) as a saddle together with \((-\sqrt{3},-\sqrt{3})\) as a local minimum and \((\sqrt{3},\sqrt{3})\) as a local maximum.

Find the maximum rate of change of \(f \! \left(x , y\right) = x^{2} {\mathrm e}^{-y}\) at \([2, 0]\) and the direction in which it occurs.

Analysis: Run 5 gives magnitude \(\sqrt{17}\) with unit direction \((4,-1)/\sqrt{17}\), while run 1 gives magnitude \(4\sqrt{5}\) with direction vector \((4,-4)\) — the scalar magnitudes and the direction vectors disagree.

Find the volume of the solid bounded above by \(z = -x^{2}-y^{2}+4\) and below by \(z = x^{2}+y^{2}\).

Analysis: The answer by run 1 is \(2\pi\), the answer by run 2 is \(4\pi\), and the answer by run 3 is \(\pi\).

Find the determinant of the 4x4 matrix \(A = \left[\begin{array}{cccc} 2 & 1 & 0 & 1 \\ 1 & 3 & 1 & 0 \\ 0 & 1 & 2 & 1 \\ 1 & 0 & 1 & 3 \end{array}\right]\).

Analysis: The answers differ in value: run 1 gives \(20\) while run 4 gives \(10\).

Find one valid QR decomposition of the matrix \(\left[\begin{array}{cc} 1 & 1 \\ 1 & 0 \\ 0 & 1 \end{array}\right]\).

Analysis: All responses are from mistral; mathematically they split because one gives a \(3\times 2\) \(Q\) with \(2\times 2\) \(R\) using a second \(Q\)-column proportional to \((1,-1,2)\), another gives a \(3\times 3\)–\(3\times 3\) pair, another gives a \(3\times 2\)–\(2\times 2\) pair with second \(Q\)-column proportional to \((2,-1,1)\), another gives a \(3\times 2\)–\(2\times 2\) pair with second \(Q\)-column proportional to \((1,-1,1)\) (and different \(R_{22}\)), and the last is only a length-2 vector rather than a \((Q,R)\) decomposition.

Find one valid singular value decomposition of the matrix \(\left[\begin{array}{cc} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{array}\right]\).

Analysis: One pair of mistral responses is the same SVD up to full-vs-thin notation with matching singular values \((\sigma_1 \approx 9.525518,\ \sigma_2 \approx 0.514300)\), while each other mistral response has different matrix factors (including different singular values and/or incompatible factor shapes), so they are not equivalent to that pair or to each other.

Diagonalize \(A = \left[\begin{array}{cc} 4 & 1 \\ 2 & 3 \end{array}\right]\) as \(A = D\), giving \(P\) and \(D\) explicitly.

Analysis: The groups differ in the eigenvalue-eigenvector pairing inside \(P\): run 1 and run 2 pair \(5\) with \([1,1]\) and \(2\) with a scalar multiple of \([-1,2]\); run 4 and run 5 pair \(5\) with \([1,2]\) and \(2\) with \([1,1]\); run 6 pairs \(5\) with \([1,1]\) and \(2\) with \([1,2]\).

Find the minimum-norm least squares solution to \(A x = b\) for \(A = \left[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right]\) and \(b = \left[\begin{array}{c} 1 \\ 1 \end{array}\right]\).

Analysis: The groups represent four distinct vectors: \(\left(-\frac{4}{3},\frac{1}{3},\frac{2}{3}\right)\), \(\left(-\frac{1}{3},\frac{2}{3},0\right)\) (with notation-only variants including a column form and an equation form), \(\left(-\frac{1}{3},\frac{1}{3},0\right)\), and \(\left(-\frac{1}{3},0,\frac{1}{3}\right)\).

Solve the Riccati equation \(\frac{d}{d x}y \! \left(x \right) = 1+x^{2}-2 x y \! \left(x \right)+y \! \left(x \right)^{2}\).

Analysis: The six answers all express \(y(x)=x+\) a different rational/transcendental correction term — involving Whittaker functions of \(x^2\), a plain exponential in \(x\), Whittaker functions of \(e^{x-C_1}\), the error function \(\operatorname{erf}(\sqrt{2}\,x)\), \(e^{-2x}\), and Bessel functions \(I_\nu(x),K_\nu(x)\) respectively — and these closed forms are not mutual reparametrizations of one another.

Solve the Riccati equation \(\frac{d}{d x}y \! \left(x \right) = y \! \left(x \right)^{2}+x^{2}\) in closed form.

Analysis: Run 3 and run 6 give the same particular Bessel-ratio solution (no arbitrary constant); run 1 has the same shape as run 5 but with the overall \(x\) factor cancelled between numerator and denominator; run 2 is a sum of Bessel functions rather than a ratio; run 4 inverts the structure with Bessel orders \(\pm 1/4\) in the numerator and \(\pm 3/4\) in the denominator; run 5 is the standard general form \(-x(C_1 J_{-3/4} + Y_{-3/4})/(C_1 J_{1/4} + Y_{1/4})\).

Solve \(\frac{d}{d x}y \! \left(x \right) = \frac{x \left(x^{2}-x -1-2 x^{2} y \! \left(x \right)+2 x^{4}\right)}{\left(x^{2}-y \! \left(x \right)\right) \left(x +1\right)}\), expressing the implicit general solution.

Analysis: The six answers have structurally distinct implicit forms — run 1 uses \((x^2-y)e^{x^2-y}\) with a polynomial in \(x\); run 2 and run 5 share the factor \((x^2-y)e^{x^2-x-y}\) but differ in the remaining term (\(x^3 e^{x^2}/3\) vs \(x^4 e^{x^2}/2\)); run 3 contains an arbitrary constant multiplying \((x+1)^2\) added to \(y\); run 4 has \(\ln(x+1)\) multiplying \(x^2\) and \(y\) with an additive constant; and run 6 uses \(\ln((x^2-y)^2(x+1)^2)\) together with \(\ln(x)\) and \(x^2\), so no two are mathematically equivalent.

Solve Bessel's equation \(x^{2} \left(\frac{d^{2}}{d x^{2}}y \! \left(x \right)\right)+x \left(\frac{d}{d x}y \! \left(x \right)\right)+\left(x^{2}-1\right) y \! \left(x \right) = 0\).

Analysis: The answer from run 2 is the full general solution \(C_1 J_1(x) + C_2 Y_1(x)\) with two independent constants, while run 1's expression \(J_1(x) + C_1 Y_1(x)\) contains only one arbitrary constant and fixes the coefficient of \(J_1(x)\) to 1.

Solve \(\frac{d^{2}}{d x^{2}}y \! \left(x \right) = \frac{x^{2} \left(\frac{d}{d x}y \! \left(x \right)\right)^{2}-2 x y \! \left(x \right) \left(\frac{d}{d x}y \! \left(x \right)\right)+y \! \left(x \right)^{2}}{x^{2}}\).

Analysis: run 1 gives two single-parameter implicit families \(C_1 x^2 + y = 0\) and \(C_1 x^2 e^{-y/x} + y = 0\); run 2 gives the two-parameter linear combination \(y = C_1 x^2 + C_2 x\); run 3 gives the two-parameter combination \(y = C_1 x \sin(\ln x) + C_2 x \cos(\ln x)\).

For the Fourier-Bessel expansion \(f \! \left(r \right) = -r^{2}+1 = \overset{\infty}{\underset{n =1}{\sum}}c_{n} J_{0}\! \left(\alpha_{n} r \right)\) on \([0, 1]\), where \(\alpha_{n}\) is the n-th positive zero of \(J_{0}\! \left(x \right)\), give the closed form for \(c_{n}\) and the numeric values of \(c_{1}\), \(c_{2}\), \(c_{3}\) to 4 decimal places.

Analysis: Run 1 and run 6 give the same closed form for \(c_n\), but their numeric values for \(c_1, c_2, c_3\) differ (\(4.2863, -1.0761, 0.5089\) vs \(4.2359, -1.0409, 0.4575\)).

Find all 6th roots of \(-64\) in polar form and identify which roots lie in the upper half-plane.

Analysis: run 2 and run 3 give six values of modulus \(2\) at arguments \(\pi/6 + k\pi/3\); run 4 gives six values of modulus \(2\) at arguments \(k\pi/3\) (offset by \(\pi/3\) from run 2 and run 3); run 1 gives six values of modulus \(2^{4/3}\) at arguments \(k\pi/3\).

Find the Laurent series expansion of \(f \! \left(z \right) = \frac{1}{z \left(z -2\right)}\) about \(z = 0\) for \(0<{| z |}<2\).

Analysis: Run 4's expansion has principal part \(-\frac{1}{2z}\) with the geometric sum \(\sum (z/2)^k\) (all negative coefficients), while run 6's expansion has principal part \(+\frac{1}{2z}\) and alternating-sign coefficients corresponding to \(\sum (-z/2)^k\).

Find the residue of \(f \! \left(z \right) = \frac{{\mathrm e}^{z}}{\left(z -\mathrm{i} \pi \right)^{2}}\) at \(z = \mathrm{i} \pi\).

Analysis: Run 1 gives \(\frac{e^{i\pi}}{2i\pi}\), run 2 gives \(-\frac{\pi e^{\pi}}{2}\), run 5 gives \(\frac{\pi e^{i\pi}}{2}\), and run 3 gives \(\frac{e^{i\pi}}{2}\), four distinct values.

Solve the polynomial equation \(x^{4}-2 x^{3}-3 x^{2}+4 x +4 = 0\) over the reals, finding all roots.

Analysis: The answer from run 1 lists explicit values \(1+\sqrt{3},\ 1-\sqrt{3},\ 2,\ -1\), while run 3 and run 2 leave the roots as the four indexed \(\mathrm{RootOf}\) expressions of the quartic.

Find the minimal polynomial of \(\alpha = \sqrt{2}+\mathit{root}_{3}\! \left(3\right)\) over \(Q\).

Analysis: The polynomials differ in their coefficients: \(-2x^3\) vs \(-6x^3\) and \(-12x\) vs \(-36x\), and the constant term \(-1\) vs \(+1\), so they are distinct polynomials.

Compute a lexicographic Gröbner basis (with \(y <x\)) of the ideal \(\langle x^{2}+y^{2}-4, x y -1\rangle\) and use it to find all complex solutions of the system.

Analysis: The answers differ in the form of the third polynomial (giving \(x\) in terms of \(y\) or a univariate polynomial in \(y\)), in whether \(x^2+y^2-4\) is kept in the basis, and in the described complex solution set (number of solutions, indexed roots vs. unindexed roots, and the substitution form \(y=1/x\) vs. \(y=4-x^2\) vs. nested \(\mathrm{RootOf}\)).

Determine the Galois group of \(x^{4}-2\) over \(Q\).

Analysis: The first group identifies the Galois group as the dihedral group \(D_4\) of order 8 (generated by a 4-cycle and a double transposition), while run 6's answer specifies a group generated by a 3-cycle and a transposition, which is \(S_4\) of order 24.

Express the elementary symmetric polynomial \(e_{3}\! \left(x_{1}, x_{2}, x_{3}\right)\) in terms of the power sums \(p_{k} = x_{1}^{k}+x_{2}^{k}+x_{3}^{k}\) for \([k = 1, 2, 3]\).

Analysis: Runs 3, 5, and 6 give \(\frac{p_1^3}{6} - \frac{p_1 p_2}{2} + \frac{p_3}{3}\); run 1 gives \(-\frac{p_1^3}{12} + \frac{p_1 p_2}{4} + \frac{p_3}{3}\); run 2 gives \(-\frac{p_1^3}{6} + \frac{p_1 p_2}{2} - \frac{p_3}{3}\) (sign-flipped relative to runs 3, 5, 6); run 4 gives \(\frac{p_1^3}{6} - \frac{p_1 p_2}{2}\) with no \(p_3\) term.

Evaluate \(\int_{0}^{\frac{\pi}{2}}\ln \! \left(\sin \! \left(x \right)\right)^{2}d x\) in closed form.

Analysis: Run 1 contains \(\zeta(2)\), \(\zeta(3)\), and mixed \(\pi^2\) terms; run 3 gives \(\frac{\pi^3}{4}+\pi\ln^2(2)\); run 5 adds a \(2\pi\zeta(3)\) term to \(\frac{\pi^3}{4}+\frac{1}{2}\pi\ln^2(2)\); run 6 gives \(\frac{\pi^3}{24}+\frac{1}{2}\pi\ln^2(2)\), with different rational coefficients on \(\pi^3\) and \(\pi\ln^2(2)\) across the four.

Find the number of binary strings of length 8 with no consecutive 1's.

Analysis: The answers differ in value: run 1 gives \(55\) while run 6 gives \(34\).

Find the number of spanning trees in the complete bipartite graph \(K_{2,3}\).

Analysis: The answer from run 5 is \(6\) while the answer from run 1 is \(12\), which are distinct numerical values.

For data \(X = [1, 2, 3, 4, 5]\) and \(Y = [ 2.1, 3.9, 4.8, 6.2, 7.0]\), fit a least squares regression line \(y = b x +a\) and compute the residual sum of squares.

Analysis: The five answers report different numerical values for the intercept \(a\), slope \(b\), and residual sum of squares.

Find the range and interquartile range of \([5, 12, 18, 23, 28, 35, 41]\) using Tukey's hinges, with the median excluded from the lower and upper halves.

Analysis: The answers give different numeric pairs for the range and IQR: run 1 reports \((18, 26)\), run 2 reports \((29, 19)\), run 4 reports \((29, 16)\), run 5 reports the hinges \([18, 36]\) with IQR \(19\), and run 3 reports \((29, 18)\).

For data \(\{1, 2, 3, 4, 5\}\), compute the least squares regression line \(y = b x +a\) and the coefficient of determination \(R^{2}\).

Analysis: The groups differ in the regression line and/or \(R^2\): run 1 gives \(y=1.6+0.7x,\ R^2=0.575\); run 2 gives the same line but \(R^2\approx 0.5352\); run 3 gives \([7/5,\,11/10]\) with \(R^2=121/170\); run 4 gives \([7/10,\,18/5]\) with \(R^2=18/37\); run 5 gives \(81/35+(6/7)x\) with \(R^2=18/35\); run 6 gives \(1.1+0.7x\) with \(R^2=0.7\).

Perform a two-tailed one-sample t-test at \(\alpha = 0.05\) for \(\mathit{H0}\) using the sample \([9, 10, 11, 12, 13]\) (assume normal population with unknown variance).

Analysis: The run 1 answer gives \(t \approx 2.449\) with \(p \approx 0.0705\), while run 2 gives \(t \approx -0.7071\) with \(p \approx 0.5190\), so both the test statistic and the p-value differ.

For \(\mathit{X\,~\,Beta(alpha=2,\,beta=3)\,}\), find \(E_{X}\), \(\mathit{Var}_{X}\), and the mode.

Analysis: The groups differ in the value given for \(\operatorname{Var}[X]\): run 3 gives \(3/50\), run 1 gives \(4/175\), run 2 gives \(6/175\), run 4 gives \(2/75\), and run 6 gives \(4/125\); the answers for \(E[X]\) also differ between run 3 (\(2/5\)) and the others (\(8/5\)).

Evaluate \(\int_{0}^{1}\frac{\ln \! \left(x \right)^{2}}{x^{2}+1}d x\) in closed form.

Analysis: Run 1 combines \(\text{Catalan}\,\pi^2/8\), \(-7\pi^3/1536\) and \(-G\pi/2\); run 2 uses \(\pi^4\) instead and only a \(\text{Catalan}\,\pi/2\) term; run 3 subtracts \(\pi G/2 + \text{Catalan}\,\pi^2/2\) from \(\pi^3/16\); run 4 and run 5 both start with \(\pi^3/16 - 7\pi\ln(2)^2/4\) but differ in the remaining \(\zeta\)-terms (run 4 has \(\pi\ln(2)\zeta(2)/2 + 7\zeta(3)/8\), run 5 has \(7\ln(2)\zeta(2)/2\)); run 6 is complex-valued with an imaginary part \(\text{Catalan}\,I/2\) added to \(\pi^3/16 - \pi\ln(2)^2/8\), and the six expressions yield six distinct numerical values.

Find the equation of the osculating circle to the curve \(y = x^{2}\) at the point \([1, 1]\).

Analysis: The groups differ in the circle's center and radius: run 1, run 2 and run 4 give center \((-1/2,\,9/4)\) with \(r^2 = 125/16\); run 3 gives center \((-1/2,\,5/4)\) with \(r^2 = 125/16\); run 5 gives center \((-1/2,\,5/2)\) with \(r^2 = 125/4\); run 6 gives center \((-1/2,\,5/4)\) with \(r^2 = 13/16\).

Evaluate \(\int_{0}^{1}\frac{\ln \! \left(1-x \right)^{2}}{x}d x\) in closed form. Express the result using \(\zeta \! \left(3\right)\).

Analysis: The answer by run 2 gives \(6\zeta(3)\), run 1 gives \(2\zeta(3)\), run 3 gives \(\zeta(3)/4\), and run 4 gives \(\zeta(3)\), all differing by numerical factors of \(\zeta(3)\).

Evaluate \(\int_{0}^{\infty}\frac{\ln \! \left(x \right)}{\left(x^{2}+1\right)^{2}}d x\) in closed form.

Analysis: run 1's answer \(-\pi^2/8\) is twice run 2's answer \(-\pi^2/16\).

Project the vector \(\langle 2, 3, 4\rangle\) onto the subspace of \(R^{3}\) spanned by \(\langle 1, 1, 1\rangle\) and \(\langle 1, 0, 0\rangle\).

Analysis: The mistral answers are pairwise nonequivalent: they represent \( \left(\frac{16}{3},\frac{14}{3},\frac{14}{3}\right) \), \( \left(\frac{11}{3},\frac{11}{3},\frac{11}{3}\right) \), \( (2,1,1) \), and \( \left(\frac{14}{3},\frac{7}{3},\frac{7}{3}\right) \), respectively.

Evaluate \(\mathrm{I}\! \left(a \right) = \int_{0}^{\infty}\frac{\cos \! \left(a x \right)}{x^{4}+1}d x\) for \(0<a\) in closed form.

Analysis: The mistral answers are pairwise nonequivalent: they have respectively \(e^{-a/\sqrt2}(\cos(a/\sqrt2)+\sin(a/\sqrt2))\), \(e^{-a/\sqrt2}\cos(a/\sqrt2)\), a mixed form simplifying to \(\cos(a/\sqrt2)+e^{-\sqrt2 a}\sin(a/\sqrt2)\), and \(e^{-a/\sqrt2}\sin(a/\sqrt2+\pi/4)\) with different overall scaling factors.

Evaluate \(\int_{0}^{2 \pi}\frac{\cos \! \left(\theta \right)}{5+4 \cos \! \left(\theta \right)}d \theta\).

Analysis: The mistral answers are pairwise nonequivalent, giving three different constants: \(\frac{\pi}{4}-\frac{\pi\sqrt{5}}{2}\), \(\frac{\pi}{4}\), and \(\frac{\pi}{2}-\frac{3\pi\sqrt{5}}{4}\).

Factor the polynomial \(x^{8}+x^{4}+1\) completely over the complex numbers.

Analysis: The four groups are mathematically different as written: they have different factor sets and different total degrees (respectively \(12,14,12,16\)), with one using explicit \(\sqrt{3}\) and \(i\), one mixing quadratic and linear complex factors, one using \(\operatorname{RootOf}\) coefficients, and one using additional \(\pm\!\left(1\pm \frac{\sqrt{3}}{3}\right)x\) quadratic factors.

Find the closed form of the infinite sum \(\overset{\infty}{\underset{n =1}{\sum}}\frac{H_{n}}{n^{3}}\), where \(H_{n} = \overset{n}{\underset{k =1}{\sum}}\frac{1}{k}\) is the n-th harmonic number.

Analysis: One mistral expression is \(\frac{7\pi^4}{720}+\frac{3}{2}\zeta(3)\), two mistral expressions are \(\frac{3}{2}\zeta(4)+c\,\zeta(2)^2\) with different \(c\) values (\(\frac12\) vs \(\frac14\)), and one mistral expression is a \(\pi,\ln 2,\zeta(3),\operatorname{Li}_4(1/2)\) combination; these are mathematically non-equivalent to each other.