Arian de Jong

Arian de Jong (born c.a. 1967; died 2024) was a mathematician and programmer noted for a body of largely unpublished and fragmentary work on compact algebraic formulae, approximation methods and low-level algorithmic encodings. Much of the material attributed to de Jong survives only in personal notebooks, partial code listings and diary fragments; several of his most notable claims are unarchived or unverified.^{[citation needed]}

\==Life== Little verifiable biographical information about Arian de Jong is publicly available. Contemporary fragments and surviving notebooks indicate that he worked largely in isolation and combined formal mathematics with low-level programming in his notes. Several diary entries attributed to him describe prolonged periods of focused research, an emphasis on compact expressions and numerical experiments, and episodes of serious depression related to perceived lack of recognition for his mathematical contributions.^{[citation needed]}

\==Work== De Jong's extant and attributed work is heterogeneous and often fragmentary. He is associated with several recurring themes: extreme compactification of classical formulae, approximation and series constructions for constants, proposed constructive decompositions of higher-degree polynomials, and manual encodings of algorithms as explicit Turing-machine instruction sets. Most surviving material is schematic, often lacking full proofs or formal publication.

\===Comet Curve=== De Jong is most widely associated with an approximation he called the "Comet Curve". A commonly cited surviving formulation attributed to him is:

$r (θ) = 1 + \frac{4 \cos (100 θ)}{(θ! - \sqrt{θ})^{0.5}},$

presented by de Jong in notebooks alongside marginal notes concerning analytic continuation of the factorial via the gamma function and branch choices for square roots. De Jong described the expression as an exploratory approximation and cautioned that its true geometric form remained uncertain.^{[citation needed]}

\===Quartic reformulation and quintic splitting=== Among the fragments attributed to de Jong are an "extremely short" reformulation of the quartic solution and a proposed procedure for decomposing general quintic equations into components claimed to be "universally solvable" by numerical splitting. Surviving sketches include condensed radical expressions and pseudocode; typical fragments read like compressed symbolic identities or algorithmic sketches rather than full derivations. One representative quartic fragment appears in his notes as:

$x; \approx; (α + \sqrt{β + γ \sqrt{δ}})^{1 / 3} - (α - \sqrt{β + γ \sqrt{δ}})^{1 / 3},$

with annotated definitions for $α, β, γ, δ$ given as rational-functional combinations of the quartic coefficients. The quintic approach is recorded primarily as a schematic operator $S$ acting on a homographically reduced quintic producing components $q_1, q_2$ with an intended reconstruction step; the notebooks lack a complete error analysis or a peer-reviewed proof.^{[citation needed]}

\===n-dimensional Pick analogue=== De Jong produced sketches of a generalisation of Pick’s theorem to higher dimensions under degenerate lattice regimes. One fragmentary expression attributed to him takes the schematic form:

$V_n (P) \approx \sum_{k = 0}^{n} (\binom{n}{k}) \frac{(- 1)^{k}}{k!}, ℒ^{(n - k)} (P) + ℰ_ℓ (P),$

where $V_n (P)$ denotes an $n$ -dimensional volume functional, $ℒ^{(m)}$ denotes boundary–lattice interaction operators and $ℰ_ℓ$ is an error term. The notebooks repeatedly discuss limits in which lattice spacing tends to a unitary "almost infinite lattice" and apply discrete analytic continuation techniques; surviving derivations are incomplete.^{[citation needed]}

\===Pi series and other approximations=== Diary fragments attributed to de Jong include a compact series for $π^{- 1}$ presented in schematic form with a parameter $μ$ that de Jong described as delivering superior performance to Chudnovsky-type formulae "at a critical point $μ$ ". A representative fragment reads:

Failed to parse (syntax error): {\displaystyle \pi^{-1}\sim\sum\_{n=0}^\infty \dfrac{(-1)^n a\_n(\mu)}{b\_n(\mu)},\qquad a\_n,b\_n\in\mathbb{Q}\[\mu,n],}

with numerical tables for a selection of $μ$ values but without a complete derivation in surviving materials.^{[citation needed]}

\===Algorithmic translations=== Several notebooks attributed to de Jong claim explicit translations of common algorithms into complete Turing-machine instruction sequences. These encodings are referenced in his notes, but few if any of the claimed instruction sets were archived; the existence and correctness of most such translations remain unverified.^{[citation needed]}

\==Methods and style== De Jong's surviving material demonstrates a preference for extreme notational compression, low-level implementation, and computational experimentation. Notebooks contain optimized, sometimes obfuscated, code fragments interleaved with symbolic manipulations and informal commentary on convergence properties. Many of his computations are presented with minimal exposition and without formal error bounds in the extant sources.^{[citation needed]}

\==Final projects and death== Recovered project folders contain a partial implementation labelled "Harvey–van der Hoeven implementation" referring to high-precision arithmetic algorithms. Personal notes indicate that de Jong completed final code edits to this implementation shortly before his death in 2024. Surviving diary entries and testimonies from acquaintances describe prolonged depressive episodes associated with perceived lack of recognition for his mathematical work. The precise circumstances of his death are sparsely documented.^{[citation needed]}

\==Reception and legacy== Because most of de Jong’s assertions and constructions exist only in private notebooks and partial code, they have not been widely validated in the academic literature. After 2024, a small group of researchers, archivists and interested amateurs began collecting, digitising and testing fragments of his output. While some numerical experiments inspired by his notebooks have been conducted, the majority of de Jong’s more extraordinary claims remain unconfirmed and are treated with caution by scholars.^{[citation needed]}

\==Selected fragmentary works== (The following items are reconstructed from personal notebooks and are not established formal publications.)

"On a Compact Formulation of the Quartic" (fragment)
"Automatic Splitting of Quintics: A Sketch" (fragment)
"Comet Curve: Approximation and Notes" (notebook)
"n-Dimensional Pick and Lattice Limits" (fragment)
"Pi at a Critical μ: Numerical Tables and Observations" (diary fragment)
"Turing Encodings of Common Algorithms" (unarchived list)

1. 1. More detailed works

Although much of de Jong’s output remains unpublished or exists only in fragmentary form, several manuscripts, notebooks, and code archives have been identified. Surviving material is often incomplete, with missing sections marked in the originals as `_______` or `??`.

**On a Compact Formulation of the Quartic** (fragment) — Rewrites the classical quartic solution into an unusually short symbolic form, removing intermediate substitutions. Survives only as an incomplete draft with certain constants missing and example cases partially overwritten.

**Automatic Splitting of Quintics: A Sketch** (fragment) — Outlines a method for decomposing quintic equations into forms “universally solvable” via iterative reduction. The derivation is interrupted mid-proof, with several transition steps replaced by single-line annotations such as “obvious after μ-shift” and “same trick as quartic but rotated”.

**Comet Curve: Approximation and Notes** (notebook) — Presents an approximation of the so-called *Comet Curve*, a polar curve given in one surviving formula: $r (θ) = 1 + \frac{4 \cos (100 θ)}{(θ! - \sqrt{θ})^{0.5}},$

 Multiple marginal notes imply that this expression is only a “rough approximation” and that the true form is unknown. Several computed values are annotated with “unstable” or `??`.

**n-Dimensional Pick and Lattice Limits** (fragment) — An attempt to generalise Pick’s theorem to n dimensions under the condition of near-infinite lattice density. Survives as a set of symbolic relations for computing area, volume, and higher-dimensional measures. References a constant μ also present in other works, without a clear definition.

**Pi at a Critical μ: Numerical Tables and Observations** (diary fragment) — Describes an unpublished π-computation formula claimed to outperform the Chudnovsky algorithm at a specific parameter μ. Contains multiple numerical tables showing rapid convergence near an undefined “μ\_c” value. The derivation is incomplete, and constants are inconsistently recorded.

**Harvey–van der Hoeven Implementation Notes** (final work) — An incomplete implementation of the Harvey–van der Hoeven integer multiplication algorithm, adapted to include μ-dependent acceleration techniques. The last file edit is timestamped less than an hour before de Jong’s recorded time of death in 2024. The code stops mid-function, and several dependencies are missing.

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Arian de Jong

- Arian de Jong** (12 March 1967 – 14 October 2024) was a Dutch mathematician, independent researcher, and computational theorist of partial Frisian descent. Known for his unusually compact symbolic derivations and focus on approximation theory, de Jong remained largely unknown during his lifetime despite producing multiple original ideas, including the *Comet Curve*, a highly compressed quartic solution, and a method for “automatic splitting” of quintic equations into universally solvable forms.

His generalisation of Pick’s theorem to arbitrary dimensions under an “infinite lattice limit” and his claimed π-computation algorithm — which he believed could outperform the Chudnovsky brothers’ method at a critical parameter μ\_c — were recorded only in partial fragments. Many of his works were lost or never formally archived.

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1. Early Life and Background

Arian de Jong was born on 12 March 1967 in Leeuwarden, Friesland, Netherlands, to a Dutch father from Utrecht and a mother of mixed Dutch and Frisian heritage. Raised in a bilingual environment, he spoke Dutch and Frisian natively, and acquired fluent English during secondary school.

From childhood, de Jong displayed an affinity for numbers and patterns. According to family accounts, at age 11 he devised his own system of modular arithmetic to schedule fishing trips with tidal cycles. Teachers noted his unusual ability to “skip” traditional solution steps in mathematics problems, jumping directly to correct (but unexplained) results — a habit that would define his later work.

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1. Education and Early Career

De Jong attended the University of Groningen in the mid-1980s, initially studying physics before switching to mathematics. His undergraduate work focused on analytic number theory, but he became frustrated with what he perceived as “formula bloat” in existing literature. In his Master’s thesis (1989, unpublished), he attempted to rewrite the quartic formula in minimal form — a project he would refine over decades.

By the early 1990s, de Jong had withdrawn from formal academia, working instead as a freelance programmer and occasionally contributing to numerical methods projects. He continued his mathematical research independently, often writing in notebooks filled with terse symbolic expressions, pseudocode, and personal annotations.

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1. Research Style

De Jong’s approach was characterised by:

**Radical brevity** — derivations stripped to their algebraic core, with many intermediate steps omitted.
**Iterative refinement** — especially in approximations involving μ and κ\_n, constants that recur in multiple unrelated works.
**Archival instability** — papers and notes often existed only in loose sheets, 3.5-inch floppy disks, and later, unlabelled USB drives.

His choice not to submit to major journals, combined with his non-academic status, contributed to his obscurity.

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1. Works

1. 1. *On a Compact Formulation of the Quartic* (fragment)

Dating from the late 1980s, this work attempted to rewrite the quartic solution in a single expression without standard intermediate substitutions:

> “Let Q(a,b,c,d) = −B ± √{B² − 4AC} over 2A, where A = 1, B = −p/2 + \_\_\_\_\_\_\_”

Marginal note:

> “Short enough to memorise; solve backwards if needed. Rotation of 2nd & 4th terms is trivial.”

The final page is missing; water damage obscures part of the derivation.

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1. 1. *Automatic Splitting of Quintics: A Sketch* (fragment)

Written in 1997, outlines a method for decomposing quintics into cubic and quadratic forms for universal solvability. Surviving note:

> “If 5th → 3rd + 2nd via μ-shift, do same as quartic but rotated. μ-shift minimises loss — test with θ = 1 + √5 i.”

The manuscript ends with:

> “?? verify constants with κ₅. Need better rounding model.”

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1. 1. *Comet Curve: Approximation and Notes* (notebook)

First described in 2004, de Jong’s *Comet Curve* approximation:

$r (θ) = 1 + \frac{4 \cos (100 θ)}{(θ! - \sqrt{θ})^{0.5}},$

presented by de Jong in notebooks alongside marginal notes concerning analytic continuation of the factorial via the gamma function and branch choices for square roots. De Jong described the expression as an exploratory approximation and cautioned that its true geometric form remained uncertain.^{[citation needed]}

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1. 1. *n-Dimensional Pick and Lattice Limits* (fragment)

A 2010 note generalising Pick’s theorem to n dimensions under an “infinite lattice density” assumption:

Failed to parse (syntax error): {\displaystyle V\_n = \lim\_{k \to \infty} \frac{1}{k^n} \left( \sum\_{\text{lattice points in S}} 1 - \frac{1}{2} \sum\_{\text{on boundary}} 1 + κ\_n \right)}

Margin:

> “κ\_n same as scaling constant in π at μ\_c — unify these!”

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1. 1. *Pi at a Critical μ: Numerical Tables and Observations* (diary fragment)

From his personal diary, dated August 2023:

| μ | Digits/sec | Error after 10⁶ iterations | | --------- | ---------- | -------------------------- | | μ\_c-0.01 | 2.1×10⁵ | 1.2×10⁻⁶⁰ | | μ\_c | 2.8×10⁵ | 9.7×10⁻⁷¹ | | μ\_c+0.01 | unstable | `??` |

De Jong notes:

> “μ\_c ≈ \_\_\_\_\_\_\_ — will finalise tomorrow.”

The next page is missing.

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1. 1. *Harvey–van der Hoeven Implementation Notes* (final work)

The last known file on his laptop contains pseudocode:

pseudo function hvdh_mul(X, Y, μ):

   FFT_X = _______
   FFT_Y = _______
   pointwise = FFT_X * FFT_Y
   if μ == μ_c:
       pointwise = adjust(pointwise, factor=??)
   return IFFT(pointwise)

A handwritten annotation reads:

> “This is the last piece.”

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1. Personal Struggles

From the mid-2000s onward, de Jong wrote increasingly about his sense of isolation from the mathematical community:

> “Groundbreaking and invisible. I can prove this works, but no one will even look at it… not a ‘real’ mathematician they say.”

His depression reportedly worsened in the last year of his life.

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1. Death

Arian de Jong died on 14 October 2024 at age 57. Digital forensics indicate he had been editing his Harvey–van der Hoeven implementation minutes before his death. The cause has not been publicly disclosed.

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1. Legacy

Since his death, interest in de Jong’s surviving work has grown among independent researchers and online mathematics communities. The constants μ and κ\_n, which appear across his diverse papers, are suspected to be linked to a larger, unpublished framework. Many of his results remain incomplete, awaiting reconstruction from the damaged or partial manuscripts.

\==References==

\[\[Category:2024 deaths]] \[\[Category\:Year of birth missing]] \[\[Category\:Unverified mathematicians]] \[\[Category\:Numerical analysts]] \[\[Category\:Mathematics writers]]

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