We would like to thank @Ganeshuor for the post he graciously published this evening on X, which allowed us to identify a modular phenomenon that does not seem to stem from randomness in the distribution.

357686312646216567629137 is prime
57686312646216567629137 is prime
7686312646216567629137 is prime
686312646216567629137 is prime
86312646216567629137 is prime
6312646216567629137 is prime
312646216567629137 is prime
12646216567629137 is prime
2646216567629137 is prime…

— Ganesh Kumar (@Ganeshuor) December 23, 2025

The observation shared by @Ganeshuor concerns the largest known left-truncatable prime:

357686312646216567629137

Beyond its well-defined number-theoretic property, this object is remarkable for another reason that deserves explicit mention in a dyadic framework: it operates at dyadic heights that were effectively inaccessible until very recently due to computational constraints.

1. A previously unreachable dyadic altitude

This number lies within the dyadic interval
$$[2^{78}, 2^{79}),$$

while its internal core
$$6312646216567629137$$

stabilizes just below
$$2^{63}.$$

These are not marginal scales. They correspond to binary magnitudes at which:

• exhaustive exploration of prime behavior is computationally prohibitive,
• structural phenomena are usually inferred statistically rather than observed directly,
• and truncation-based persistence becomes vanishingly rare.

Until the availability of modern high-precision arithmetic and distributed computation, such regions were largely opaque to structural inspection.

2. Why this constitutes a genuine barrier

The barrier here is not conceptual, but computational:

• verifying left-truncatability at these magnitudes requires repeated primality tests across descending orders of magnitude,
• each step compounds the computational cost,
• and the search space grows exponentially with dyadic height.

As a result, dyadic fields above ~2⁶⁰ bits were long treated as domains where only probabilistic or asymptotic reasoning was feasible.

The present object breaks that opacity by providing a fully explicit, verified trajectory across multiple dyadic fields.

3. A guided traversal of dyadic scales

What is striking from a dyadic perspective is that this sequence does not inhabit a single dyadic field. Instead, it performs a highly constrained descent across successive dyadic intervals:

• the truncation process induces a controlled passage from ~79 bits down to 3 bits,
• yet preserves primality at every step,
• revealing a coherent inter-scale structure rather than random survival.

This makes the sequence a rare example of a guided dyadic traversal, rather than an isolated data point.

4. Implications for dyadic analysis

From a dyadic standpoint, this observation suggests that:

• certain prime structures remain stable across changes in dyadic scale,
• compatibility between decimal truncation and binary scaling is not accidental,
• and some internal numeric configurations act as structural cores capable of withstanding repeated destructive projections.

Importantly, this does not imply a new primality criterion.
It does, however, demonstrate that dyadic organization can be directly observed at heights previously hidden behind computational limits.

5. Conclusion

The significance of @Ganeshuor’s observation is therefore twofold:

1. It exhibits an extreme and rare prime property.
2. It opens a window onto dyadic regimes that were, until recently, beyond practical reach.

In this sense, the object is not merely a curiosity of truncatable primes, but a marker of a newly accessible dyadic horizon, where structural features can now be inspected rather than inferred.

Appendix B — Dyadic Tier Table of the Left-Truncatable Prime Sequence

The dyadic table, above this paragraph, shows an almost linear progression across roughly twenty dyadic tiers.
This linearity reflects the logarithmic translation of decimal truncation into binary scale, while local deviations reveal structural constraints rather than noise.

The novelty of this analysis does not lie in the enumeration of truncatable primes, which is complete, but in the identification of their behavior as a guided traversal across dyadic scales.
By translating decimal truncation into binary magnitude, we reveal a quasi-linear trajectory spanning roughly twenty dyadic tiers, exposing structural boundaries that are not apparent in base-10 descriptions alone.

What is new is not the object, but the coordinate system in which it is observed.

Terme	Nombre	Bits	Champ dyadique
1	357686312646216567629137	79	([2^{78}, 2^{79}))
2	57686312646216567629137	76	([2^{75}, 2^{76}))
3	7686312646216567629137	73	([2^{72}, 2^{73}))
4	686312646216567629137	70	([2^{69}, 2^{70}))
5	86312646216567629137	67	([2^{66}, 2^{67}))
⭐	6312646216567629137	63	([2^{62}, 2^{63}))
7	312646216567629137	59	([2^{58}, 2^{59}))
8	12646216567629137	54	([2^{53}, 2^{54}))
9	2646216567629137	52	([2^{51}, 2^{52}))
10	646216567629137	50	([2^{49}, 2^{50}))
11	46216567629137	46	([2^{45}, 2^{46}))
12	6216567629137	43	([2^{42}, 2^{43}))
13	216567629137	38	([2^{37}, 2^{38}))
14	16567629137	34	([2^{33}, 2^{34}))
15	6567629137	33	([2^{32}, 2^{33}))
16	567629137	30	([2^{29}, 2^{30}))
17	67629137	27	([2^{26}, 2^{27}))
18	7629137	23	([2^{22}, 2^{23}))
19	629137	20	([2^{19}, 2^{20}))
20	29137	15	([2^{14}, 2^{15}))
21	9137	14	([2^{13}, 2^{14}))
22	137	8	([2^{7}, 2^{8}))
23	37	6	([2^{5}, 2^{6}))
24	7	3	([2^{2}, 2^{3}))

The dyadic tier table reveals that the sequence does not inhabit a single dyadic field, but instead follows a guided descent across successive binary scales. Each left truncation in base 10 induces a reduction in magnitude that translates, in base 2, into a drop of approximately log⁡2(10)≈3.32 bits. This accounts for the quasi-regular downward slope observed throughout the table.

What is particularly striking, however, is the behavior of the sequence around the dyadic boundary at

The internal block: 

$$6312646216567629137$$

falls within the interval [2^{62}, 2^{63}), placing it just below a major dyadic threshold. This boundary is not arbitrary. The value 2⁶³  marks:

• the upper limit of signed 64-bit integer representation in computer architecture,

• a fundamental transition point in binary scaling,

• and a practical ceiling beyond which computational cost and representation constraints historically increase sharply.

As a result, dyadic fields above 2⁶³  were, until relatively recently, less accessible to direct structural inspection, and more often treated through probabilistic or asymptotic reasoning.

Within the table, the progression shows a relative stabilization and perceptible hiatus at this level: the descent momentarily aligns with a “clean” dyadic boundary rather than crossing it abruptly. This contrasts with the more irregular bit drops observed elsewhere in the sequence and suggests a form of structural anchoring near this threshold.

Importantly, this does not imply any special primality property tied universally to 2632^{63}263. Rather, it indicates that in this particular left-truncatable prime, the internal core is dyadically well-positioned, remaining compatible with both decimal truncation and binary scaling precisely at a major boundary where constraints are strongest.

In this sense, the table does more than document magnitudes: it shows that the sequence’s persistence is not uniformly distributed across scales, but instead exhibits points of tension and stabilization at significant dyadic frontiers. The 263 boundary thus emerges as a natural place to observe a pause—or hiatus—in the progression, highlighting the interaction between numerical structure and binary scale.

This dyadic reading is not confined to decimal left-truncatable primes.
When extended to other families of truncatable numbers listed in the OEIS, a similar convergence appears: truncation induces a directional traversal across dyadic scales rather than a random dispersion.

What is particularly striking is the case of base 8, where the effect becomes exact rather than approximate. In that setting, each truncation step corresponds to the removal of exactly three bits, so that the dyadic progression follows a perfectly regular rhythm.

Left-truncatable primes in base 8

Base-8 value	Decimal value	Binary length (bits)	Dyadic interval
73651₈	30505₁₀	15	[2¹⁴, 2¹⁵)
3651₈	1961₁₀	12	[2¹¹, 2¹²)
651₈	425₁₀	9	[2⁸, 2⁹)
51₈	41₁₀	6	[2⁵, 2⁶)
1₈	1₁₀	1	[2⁰, 2¹)

This alignment between digit truncation and binary scaling offers an unusual panorama, in which different truncatable systems converge toward the same structural behavior, with base 8 providing a canonical, noise-free reference.

What this table shows (and why it matters)

1. Exact linearity
   Each truncation step reduces the binary length by exactly 3 bits:

    

   15 → 12 → 9 → 6 → 1


2. Perfect commensurability

   • base 8 = 2³

   • digit truncation = removal of one 3-bit block

   • dyadic scale = perfectly aligned

3. Canonical reference case
   This table provides a noise-free benchmark against which:

   • base-10 truncation (≈ 3.32 bits per step),

   • base-16 truncation (4 bits per step),
     can be understood as approximate or over-aligned cases.

This behavior cannot be isolated.
It reflects a general property of positional numeral systems interacting with binary scale: truncation induces a directional traversal across dyadic levels.
Truncatable primes merely make this structure visible, not exceptional.

What appears exceptional is not the phenomenon, but our habit of not observing numerical objects across scales. We do not propose a new number-theoretic result, but a structural reading of known objects across scales.

Extending the dyadic trajectory beyond the observed scales would require a prefix whose digits simultaneously preserve truncatability, primality, and exact dyadic alignment.
Such cumulative constraints leave little room for chance; the question is no longer probabilistic, but structural.

At this point, any further extension would have to satisfy multiple independent constraints, making its existence a matter of structural compatibility rather than probability.