tag: mathematics
39 proofs
0.999... (with 9s repeating forever) is strictly less than 1.
2026 being divisible by 2, 3, 7, 11, 13, 17 proves it has "hidden perfect number properties."
2026 has no square root
2026 is both a "happy number" and mathematically "perfect," proving the year is cosmically special.
641 divides \(2^{32} + 1\) exactly.
A man on TikTok has solved the Riemann Hypothesis after one week of work.
Sources: Wikipedia: Riemann Hypothesis, Wikipedia: Millennium Prize Problems, Clay Mathematics Institute: Riemann Hypothesis (Millennium Prize)
A mathematical model proves the world will end on a specific day in 2026.
Sources: Wikipedia: Future of Earth (synthesizing peer-reviewed astrophysics), LADbible: Chilling mathematical equation predicted world end date (Jan 2026), Live Science: How Much Longer Can Earth Support Life?
A real number is rational if and only if its decimal expansion is eventually periodic.
A venture capital fund investing $50 million in 25 startups at equal sizes requires at least one 50x return and two 10x returns to achieve a 3x gross multiple.
An object moving at exactly 0.95c relative to a stationary observer experiences a Lorentz factor γ greater than 3.2.
Computers had a hidden "math glitch" preventing division by zero until it was patched in 2026.
Sources: Wikipedia: IEEE 754 floating-point standard, Wikipedia: Division by zero, OSDev Wiki: x86 Processor Exceptions
Consider a sector with \(N \geq 2\) symmetric firms, each endowed with L task-positions. Each firm i chooses an automation rate \(\alpha_i\) in [0,1], paying wage w per human-staffed task and cost c per automated task, with integration friction cost \((k/2) \cdot L \cdot \alpha_i^2\) where \(k > 0\). Aggregate sectoral demand is \(D = A + \lambda \cdot w \cdot L \cdot [N - (1-\eta) \sum_j \alpha_j]\), where \(A > 0\) is exogenous demand, \(\lambda \in (0,1]\) is workers' marginal propensity to consume from wages, and \(\eta \in [0,1)\) is the fraction of displaced wage income recovered through reemployment. Each firm's revenue is \(D/N\) (equal market shares). Define \(s = w - c > 0\) and \(\ell = \lambda(1-\eta)w > 0\). Each firm i maximizes \(\pi_i = D/N - wL(1-\alpha_i) - cL\alpha_i - (k/2)L\alpha_i^2\). The Nash equilibrium automation rate is \(\alpha^{NE} = (s - \ell/N)/k\). The cooperative optimum is \(\alpha^{CO} = (s - \ell)/k\). The difference \(\alpha^{NE} - \alpha^{CO} = \ell \cdot (1 - 1/N)/k\) is strictly positive.
For a correctly specified discrete-time spike train model with conditional spike probabilities p_k, define A_m = sum_{k=tau_{m-1}+1}^{tau_m-1} -log(1-p_k) and R_m = A_m - log(1 - U_m * p_{tau_m}), where U_m ~ Uniform(0,1) independently. Then R_m are i.i.d. Exp(1), equivalently Z_m = 1 - exp(-R_m) are i.i.d. Uniform(0,1).
Every elementary function that appears on a standard scientific calculator — including \(+\), \(\times\), \(\div\), exponentiation \(x^y\), \(\sin\), \(\cos\), \(\tan\), \(\sqrt{x}\), \(\log_{10}\), \(\pi\), \(e\), \(i\), and their compositions and inverses — can be realised as a finite binary tree of the operator \(\mathrm{eml}(a, b) = e^{a} - \ln b\) whose leaves are the constant \(1\) and the input variables. Each construction is verified to machine precision at multiple test points on its natural domain.
The binary operator eml is defined by the expression \(\text{eml}(a, b) = \exp(a) - \ln(b)\). There exists a finite binary tree consisting solely of eml operations, whose 9 leaves are drawn from \(\{1, x, y\}\), such that the tree evaluates exactly to \(x \times y\). The tree has K = 17 tokens (8 eml operations and 9 leaves), and the identity holds for all complex \(x\) and \(y\) (in the algebraic setting where \(\ln \circ \exp\) is the identity).
The binary operator eml is defined by the expression \(\text{eml}(a, b) = \exp(a) - \ln(b)\). There exists a finite binary tree consisting solely of eml operations, whose 10 leaves are drawn from \(\{1, x, y\}\), such that the tree evaluates exactly to \(x + y\). The tree has K = 19 tokens (9 eml operations and 10 leaves), and the identity holds for all real \(x\) and \(y\) (and formally for all complex \(x, y\) in the algebraic setting where \(\ln \circ \exp\) is the identity).
Using only the eml operator applied to the constant 1 (and allowing complex intermediates), there exist finite expressions that evaluate exactly to the mathematical constant \(\pi\) and to the imaginary unit \(i\). These expressions can be verified by symbolic simplification or by numerical evaluation that matches the known values \(\pi \approx 3.1415926535\ldots\) and \(i^2 = -1\) to machine precision.
The binary operator eml is defined by the expression \(\text{eml}(a, b) = \exp(a) - \ln(b)\) (where exp is the exponential function and ln is the principal branch of the natural logarithm). For every real \(x > 0\), the nested expression \(\text{eml}(1, \text{eml}(\text{eml}(1, x), 1))\) equals the natural logarithm \(\ln(x)\).
Fewer than 20 percent of the positive integers from 1 to 1000 are prime.
For any triangle with sides a, b, c where \(a^2 + b^2 = c^2\), the angle opposite c is exactly 90 degrees, AND the converse also holds.
If a company raises $5 million in Series A at a $25 million post-money valuation and exits at $500 million seven years later, the annualized return to the Series A investor exceeds 40% before dilution.
Let (X,Y) ~ p(x,y). For each contrastive training instance, sample one positive Y_1 ~ p(y|X) and N-1 negatives Y_2,...,Y_N iid ~ p(y), conditionally independent given X. Let i* be the index of the positive, uniformly distributed over {1,...,N}. For any measurable scoring function s(x,y), define L_N(s) = - E[ log( exp(s(X,Y_{i*})) / sum_j exp(s(X,Y_j)) ) ]. Then log N - L_N(s) is a lower bound on I(X;Y). The Bayes-optimal score is s*(x,y) = log(p(y|x)/p(y)) + c(x), where c(x) is arbitrary. Under the standard multi-sample setup, the resulting InfoNCE lower bound tightens as N increases.
Presenting empirical spreadsheet observations as universal theorems violates the hypothetico-deductive method as defined by mainstream philosophy of science.
Sources: Encyclopaedia Britannica — criterion of falsifiability, Stanford Encyclopedia of Philosophy — scientific method, Catalog of Bias — data-dredging bias
Consider a spike-train encoding model where spikes are generated by an inhomogeneous Poisson process with intensity lambda_t = f(eta_t), eta_t = x_t^T beta + h_t^T gamma + b, with convex parameter space for theta = (beta, gamma, b). If f is positive, convex, and log-concave, then the log-likelihood is concave in theta. Therefore every local maximum is global, ML fitting is a convex optimization problem, and the same holds for MAP inference under any log-concave prior on theta.
"Thank God 2026 has no square root" is mathematically meaningful
The 100000th prime number is exactly 1299709.
The adult human brain has approximately 86 billion neurons and an average of 7,000 synapses per neuron, resulting in a total synaptic count exceeding 6 × 10^14.
Sources: Frontiers in Human Neuroscience, Herculano-Houzel 2009 (PubMed Central), UCLA Brain Research Institute, Brain Facts, BioNumbers BNID 112055, Harvard Medical School (citing Drachman 2005, Neurology)
The binary operator defined by \(\text{eml}(a, b) = \exp(a) - \ln(b)\) satisfies \(\text{eml}(x, 1) = \exp(x)\) for every complex number x.
The binary operator eml is defined by the expression \(\text{eml}(a, b) = \exp(a) - \ln(b)\) (where exp is the exponential function and ln is the principal branch of the natural logarithm). The expression \(\text{eml}(1, 1)\) equals the base of the natural logarithm \(e\).
The Goldbach conjecture holds for every even integer greater than 2.
The infinite sum from n=1 to infinity of \(1/n^2\) equals exactly \(\pi^2/6\)
The integer 1 is a prime number.
The pattern-matching limitations identified in GSM-NoOp are practically surmountable when LLMs are allowed to offload formal reasoning steps to code execution.
Sources: Mirzadeh et al., GSM-Symbolic (ICLR 2025), EmergentMind GSM-Symbolic Analysis, AppleInsider coverage of GSM-Symbolic research +5 more
The positive root of x² - x - 1 = 0 is exactly (1 + √5)/2
The Schwarzschild radius of the Sun calculated via rs = 2GM/c² with 2022 CODATA values for G, solar mass, and c lies strictly between 2.95 km and 2.96 km.
Sources: NIST 2022 CODATA Recommended Values, IAU 2015 Resolution B3 (Mamajek et al. 2015, arXiv:1510.07674), Wikipedia — Solar mass (TCG-compatible GM☉ estimate)
The St. Petersburg paradox has a finite expected value that a rational person should be willing to pay.
Sources: Wikipedia: St. Petersburg paradox, Stanford Encyclopedia of Philosophy: St. Petersburg Paradox
The viral expression "8 ÷ 2(2+2)" equals 1
Sources: Wikipedia — Order of operations