Gilbreath's Conjecture

Gilbreath's Conjecture

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

Let $p_j$ be the $j$th prime. Define $d_j^{(0)}=p_j$ and $d_j^{(r+1)}=|d_{j+1}^{(r)}-d_j^{(r)}|$ for $j\geq1,r\geq0$. The assertion is $d_1^{(r)}=1$ for every integer $r\geq1$. The initial prime row, whose first entry is 2, is not one of the asserted difference rows. [S1] [S2]

Statement recorded in the supplied source.

Starting with the sequence of primes and repeatedly taking absolute differences of consecutive terms, the first term of each row is always 1. [S1]

Short English statement

After repeatedly replacing adjacent primes, and then adjacent entries, by their absolute differences, must every newly formed row begin with one?

Research attempt

Scope and literature check.

The required assertion is at every positive difference depth for the actual ordered primes. Chase's random analogue [E1], and the 2026 Chase--Hunter--Tao manuscript [E2], concern random models and related deterministic obstructions; neither abstract asserts a proof of the prime statement. Plouffe's finite verification [S3] is likewise not an infinite conclusion. Those source statements were inspected on 1 October 2026. Here the route is an exact analysis of the left boundary and of the binary region of a difference triangle.

Parity is preserved, but is weaker than the conjecture.

For any integers $u,v$, $|u-v|\equiv u+v\pmod2$. Since the first prime gap is one and all subsequent prime gaps are even, induction gives \[ d_1^{(r)}\text{ odd},\qquad d_j^{(r)}\text{ even for }j\ge2, \qquad r\ge1. \] Indeed the new left entry is the difference of an odd and an even entry, and every tail entry is the difference of two even entries. This rules out zero on the left, but it leaves $3,5,7,\ldots$ possible. An arbitrary increasing odd tail does not remove that possibility: the sequence $2,3,7$ has difference rows $(1,4)$ and $(3)$. These are primes but are not consecutive primes; the example pinpoints why parity or primality of individual entries alone cannot prove the target.

An exact boundary criterion.

Suppose that $d_1^{(r)}=1$. The next entry equals one if and only if \[ |d_2^{(r)}-1|=1 \quad\Longleftrightarrow\quad d_2^{(r)}\in\{0,2\}. \] It follows, by induction starting from $d_1^{(1)}=1$, that the full conjecture is equivalent to \[ d_2^{(r)}\in\{0,2\}\quad\hbox{for every }r\ge1. \] This is not a new estimate: it replaces a boundary value by an exact condition on the column immediately to its right. It does show precisely what would be needed to continue a verified initial segment. At the first failure depth $r+1$, the first column was one at depth $r$, and the second entry was an even integer at least four.

An invariant region and a finite certificate.

A finite row $(1,e_2,\ldots,e_L)$ with every $e_j\in\{0,2\}$ has a successor of the same type: $|1-e_2|=1$ and $|e_{j+1}-e_j|\in\{0,2\}$. Repeating gives a left entry of one for every remaining row down to the final singleton. Therefore a finite verification can stop its expensive magnitude tracking once it reaches such a row. The argument certifies the rest of that finite triangle. It does not certify rows requiring primes absent from the input.

The same reasoning would work for a truly infinite row of this type, if one could prove its existence at some fixed depth. Establishing it for every finite prefix with a depth that varies with the prefix is different: those depths may tend to infinity. An interchange of those quantifiers would be an unproved step. The direct boundary criterion does not require a common stabilization row, so such a common row is only one possible sufficient approach, not a necessary form of a proof.

Inside the binary region: an exact linear model.

Write $e_j=2b_j$ with $b_j\in\{0,1\}$. On such a row, \[ \frac{|e_{j+1}-e_j|}{2} =b_j+b_{j+1}\pmod2, \] and the residue uniquely specifies a value in $\{0,1\}$. If $D$ is this binary difference operator, induction and Pascal's identity give \[ (D^s b)_j=\sum_{t=0}^s\binom{s}{t}b_{j+t}\pmod2. \] In particular $(D^{2^h}b)_j=b_j+b_{j+2^h}\pmod2$, because in characteristic two $(1+X)^{2^h}=1+X^{2^h}$. Thus large blocks of already-binary rows can be advanced using sparse linear operations. All dependencies remain within the original interval $[j,j+s]$.

For a general nonnegative integer row the same identity is valid only for parity. It gives no upper bound on magnitude. For example a tail entry equal to four and one equal to zero have the same residue modulo two after division by two, but their absolute differences with two are both two while other neighbors can distinguish their magnitude. The linear formula therefore cannot be substituted for the integer dynamics before entry into the binary region. This is the central obstruction to turning a fast finite algorithm into a universal magnitude bound.

Why deleting absolute values loses the problem.

Signed finite differences have the familiar alternating binomial formula, but signs at successive nodes here depend on earlier values. Already the initial primes $2,3,5,7$ give absolute-difference rows $(1,2,2)$, $(1,0)$, $(1)$, whereas the third signed forward difference is $-1$. Taking the absolute value of the final signed difference happens to work in this tiny case, but intermediate absolute values cannot in general be moved outside an iterated difference. For example the nonnegative row $(0,2,1,3)$ gives absolute rows $(2,1,2)$, $(1,1)$, $(0)$, while its third signed forward difference is six. No linear binomial estimate controls the desired entry merely by equality with that signed expression.

Executed experiment.

The script uses the first $1000$ consecutive primes, ending at $7919$, and computes all $999$ difference rows with integer arithmetic. Every first entry is one. The first row whose entire remaining tail lies in $\{0,2\}$ is row $35$; the invariant-region proof certifies the remaining finite triangle, and direct computation checks it too. See checks.py and results.json in _Local/Erdos_problems/UnsolvedMath/attempt_audit_2026_10_01/number_theory/. The check covers depths through $999$ only; the input bound is a number of primes, not a claim about all primes below a literature record bound.

Remaining gap and outcome.

The exact unresolved task is to exclude $d_2^{(r)}\ge4$ at every depth for the consecutive-prime input. The parity identity and binary stabilization lemma do not supply that bound. Random-model results require hypotheses about their input process which are not verified for the prime gaps here. Outcome: UNRESOLVED. This attempt provides an exact boundary reduction and a finite certificate mechanism, with no claim of a new theorem resolving the conjecture.

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 28 (NT-009), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 Zachary Chase, A random analogue of Gilbreath's conjecture, Mathematische Annalen 388 $2024$, 2611-2625,

  \nolinkurl{doi:10.1007/s00208-023-02579-w.}

  https://link.springer.com/article/10.1007/s00208-023-02579-w. Bibliographic information from the supplied record.

• S3 Simon Plouffe, Verification of Gilbraith's conjecture up to 10^14, arXiv:2510.06688 $2025$. https://arxiv.org/abs/2510.06688. Bibliographic information from the supplied record.

• E1 Zachary Chase, A random analogue of Gilbreath's conjecture, arXiv:2005.00530, later published in Mathematische Annalen. https://arxiv.org/abs/2005.00530. Abstract inspected 1 October 2026; random-input theorem, not a proof for the primes.
• E2 Zachary Chase, Zach Hunter, and Terence Tao, Gilbreath's conjecture: a Cramér random model and a deterministic analysis, arXiv:2607.08712 $2026$. https://arxiv.org/abs/2607.08712. Abstract inspected 1 October 2026; the full proof was not independently audited in this attempt.