A digital anomaly is a quadruple (x,y,B,k) satisfying the paper's definition. In the case k=2, the conjecture concerns the complete list of such quadruples. Digital-anomaly conjecture for k=2. The only digital anomalies (x,y,B,k) with k=2 are (x,y,B,k)=(18,4,6,2)quadandquad(1323,36,42,2). The conjecture is proposed because the abc conjecture does not appear to yield the required positive exponent for the k=2 case. Its resolution status is not specified in the source.

The conjecture on digital anomalies with exponent two

From papers

A digital anomaly is a quadruple $(x,y,B,k)$ satisfying the paper's definition. In the case $k=2$ , the conjecture concerns the complete list of such quadruples.

Digital-anomaly conjecture for $k=2$ . The only digital anomalies $(x,y,B,k)$ with $k=2$ are

(x,y,B,k)=(18,4,6,2)\quad\text{and}\quad(1323,36,42,2).

The conjecture is proposed because the $abc$ conjecture does not appear to yield the required positive exponent for the $k=2$ case. Its resolution status is not specified in the source.

Progress summary

Open

The conjecture remains open: the paper lists two examples but gives neither a proof that they are the only ones nor a counterexample.

The conjecture asserts that the two listed quadruples exhaust digital anomalies with exponent $k=2$ . It appears as Conjecture 4.6 in Samer Seraj’s paper, published in December 2025.

Known results

Assuming the $abc$ conjecture, the paper proves finiteness for each fixed $k \ge 3$ ; this argument does not address $k=2$ .

December 2025 formulation

Seraj explicitly presents the $k=2$ classification as a conjecture and says the $abc$ approach does not appear feasible because the required positive exponent is unavailable. No retrieved source reports a proof, counterexample, verification, or retraction.

Current status (as of August 2026): the two listed examples are recorded, but the $k=2$ classification remains unproved and no verified counterexample is reported.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

Samer Seraj, “Diophantine Analysis of a Digital Anomaly”, arXiv:2512.06056 (2025).

diophantine-equations number-theory

1 week ago

The proposed two-element list is false. In fact, there are infinitely many digital anomalies with exponent $k=2$ .

For each integer $m\ge1$ , define positive integers $a_m,b_m$ by

b_m+a_m\sqrt2=(3+2\sqrt2)^m.

Taking norms gives Pell's equation

b_m^2-2a_m^2=1.

Every $a_m$ is even: $a_1=2$ , and

a_{m+1}=2b_m+3a_m.

Now set

y_m=a_m^2,\qquad B_m=a_mb_m,\qquad x_m=\frac{a_m^2b_m^2}{2} =\frac{B_m^2}{2}.

These are positive integers, and the defining digital-anomaly equation holds:

\frac{x_m}{y_m} =\frac{b_m^2}{2} =a_m^2+\frac12 =y_m+\frac{x_m}{B_m^2}.

Moreover, since $B_m\ge6$ ,

B_m\le x_m=\frac{B_m^2}{2}<B_m^2.

Thus $x_m$ has exactly two base- $B_m$ digits, and

(x_m,y_m,B_m,2)

is a digital anomaly for every $m\ge1$ . The Pell solutions are strictly increasing, so these anomalies are pairwise distinct.

For $m=1$ , this construction recovers the known example

(18,4,6,2).

For $m=2$ , it produces the unlisted counterexample

\boxed{(20808,144,204,2).}

Indeed,

\frac{20808}{144} =144+\frac12 =144+\frac{20808}{204^2}, \qquad 204\le20808<204^2=41616.

Consequently, the conjectured classification omits infinitely many solutions.

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Shivam Patel · 23 hours ago