Conjecture. For every prime $p>2$ there are
$q, r \in \{1\}\cup\mathbb{P}$ with $q<p$ such that
$p = 2q + r$ or $p = 2q - r$. Since $r$ decomposes too, every
prime unfolds into a binary tree whose leaves are all $1$.
Stronger C2-Additive form: $p = 2q + r$ with $q < p/2$, giving one canonical tree, no ± ambiguity.
Strictly stronger than Lemoine's conjecture (1895) on the primes: proving C2 gives Lemoine for primes as a corollary.
Verified for all 1,008,309,544 primes (up to ≈ 2·10¹⁰) with zero counterexamples, using a two-stage CUDA pipeline on an RTX 5070 at ≈150M primes/s.
Open question: does iterating $2q + r$ from $S_0=\{1,2\}$ generate exactly the primes? If so, it is a definition of primality that doesn't go through factorisation.
Paul Lockhart (A Mathematician's Lament) judged it “possibly even more difficult” than classical problems such as $p = n^2+1$.
Figure 1. The decomposition tree of any prime. Solid edges go
to $q$ (doubled), dashed edges to $r$. Type a prime or roll a random
one; hover a node to see its equation.
PreprintJune – Sept 2025
Discrete Square Residual Structures (DSRS): every integer reveals its own connection to π
shows a sharp dichotomy (checked up to $N=10^6$, $\mu\le 70{,}000$):
$P(\mu)\to\pi/2$ when $\mu\not\equiv 0 \pmod 4$, and $P(\mu)\to 1$ when
$4\mid\mu$ or $\mu=1$.
Every non-multiple of 4 reduces to $\mu=2$ (a floor/ceiling encoding of the Wallis product) by residual cancellation.
Figure 2. Partial products $P_N(\mu)$ for $N$ up to $4\cdot10^4$,
computed live in your browser. Slide $\mu$ through a multiple of 4 and
watch the limit jump from $\pi/2$ to $1$.
Preprint · ongoingApr 2025 – present
Fibonacci Numbers from Pascal Rows: A Ternary Coefficient Approach
For which signs $c_i\in\{-1,0,1\}$ does
$\sum_{i=0}^{n-1} c_i\binom{n-1}{i} = F_n$? A row of Pascal's
triangle, a choice of add / subtract / skip for each entry, and a
Fibonacci number as the target.
Six complete enumeration strategies with CUDA acceleration, exhaustive up to $3^{22}$ (31.4 billion) combinations.
A greedy $O(n)$ heuristic for the $O(3^n)$ space, built on palindromic symmetry and centre-dominance.
I have since found cases where the greedy rule fails; I'm now working out exactly when it holds.
Arrange the iterated forward differences $\Delta^{\ell}(x^p)$ as a
family of triangles indexed by the layer $\ell$. Layer 0 is Pascal's
triangle; higher layers keep a Pascal-like organisation.
A closed-form generator $C_{p,\ell}(v)$ produces single coefficients directly (no expansion, no Stirling-basis change), with $O(\ell)$ access when $p\gg\ell$.
Unifies five OEIS sequences (A259569, my own closed form, plus A001117, A000918, A000919, A000920) as fixed-layer projections of one structure.
Connects to Voigt (1901, Konformanten) and Piza (1947, D-coefficients), whose values appear in layers 2 and 3 but were never organised or unified.
Figure 4. Coefficients of $\Delta^{\ell}(x^p)$ (layer 0: $(x+1)^p$), highest power
first, computed exactly with BigInt. Raise $\ell$: the constant
terms (red) become $\ell!\,S(p,\ell)$, the OEIS sequences named in
the readout. Hover any cell.
Conference talkAug 2022 – Dec 2024
Probability in Regular 2-Polytopes
Indian Mathematical Society Conference, MIT-WPU Pune · 25 December 2024
Drop a point (a 0-polytope) uniformly inside a regular polygon. Is it
closer to the centroid or to the boundary? I proposed a
periodic cotangent function to model how this probability
depends on the polygon.
Independent work, 2022–2024; presented at the IMS conference with feedback from working mathematicians.
Written paper in progress; the full argument is in the talk video.
Figure 5. Monte Carlo: coloured points are nearer the centroid
than any edge. The “exact” value is the area under the $n$ parabolic arcs,
$\frac{t+t^3/3}{2\tan(\pi/n)}$ with $t=\tan\frac{\pi}{2n}$, which tends
to $1/4$ (the disc) as $n\to\infty$.
Working paperIn progress
Dimensional Primality: A Geometric Hierarchy of Integers via Discrete Hyper-Rectangles
Treat $N$ as the volume of a box whose integer sides are all
greater than 1. The highest dimension in which $N$ can be built as
such a genuine hyper-rectangle is exactly $\Omega(N)$, the
number of its prime factors counted with multiplicity.
A strictly nested hierarchy $D_1 \supsetneq D_2 \supsetneq D_3 \supsetneq\cdots$ with $D_n=\{N:\Omega(N)\ge n\}$ and $\bigcap_n D_n=\varnothing$.
$n$-dimensional primes are the integers outside $D_n$: $1$ together with the $k$-almost primes for $k\le n-1$. The case $n=2$ gives back the ordinary primes.
Forced collapse: written in more than $\Omega(N)$ dimensions, at least $n-\Omega(N)$ sides must equal $1$.
The smallest member of $D_n$ is $2^n$, reached only by the cube $(2,\dots,2)$, the point where each new dimension first becomes possible.
Figure 6. Top: $N$ built as an $n$-dimensional box (dimensions
above 3 appear as stacked copies). Raise $n$ past $\Omega(N)$ and the box
collapses: sides of $1$ appear. Bottom: $\Omega(k)$ for $k\le 64$; bars
above the dashed line are in $D_n$, outlined bars are the thresholds
$2^n$. Click a bar to draw that number.