Boyd’s Conjecture: Mahler Measures & Arithmetic Patterns

• Boyd's Conjecture is a multifaceted set of hypotheses linking experimental arithmetic patterns, including rational relations between Mahler measures and special L-values.
• It predicts that regulator integrals arising from two-variable Laurent polynomials on elliptic and genus‑2 curves are rationally related to derivatives of L‑functions.
• It also explores combinatorial and distributional aspects in beta‑expansions, Salem-Pisot number phenomena, and p‑adic behavior in harmonic numbers.

Searching arXiv for papers on "Boyd's conjecture" across the main mathematical contexts represented in the provided material. In the cited literature, “Boyd’s conjecture” does not designate a single statement uniformly across mathematics. It most often denotes a family of conjectural phenomena proposed by Boyd in which experimentally observed algebraic quantities are expected to coincide with special values of arithmetic invariants, especially Mahler measures and derivatives of elliptic $L$-functions. In other contexts, the same label refers to conjectures on accumulation points of Salem numbers, cyclotomic co-factors in $\beta$-expansions for regular Pisot numbers, asymptotic distributions of conjugates of extremal algebraic integers, and $p$-divisibility properties of harmonic numbers (Meemark et al., 2019, Amara, 24 Sep 2025, Panju, 2011, Carofiglio et al., 19 Mar 2025).

1. Main meanings of the term in current literature

The most developed usage is the Mahler-measure formulation, where Boyd observed numerically that for many two-variable Laurent polynomials $P$ defining elliptic curves, the logarithmic Mahler measure

$$m(P)=\frac{1}{(2\pi)^n}\int_{0}^{2\pi}\cdots\int_{0}^{2\pi}\log\big|P(e^{i\theta_1},\ldots,e^{i\theta_n})\big|\,d\theta_1\cdots d\theta_n$$

is rationally related to $L'(E,0)$ for the associated elliptic curve $E$. Closely related genus-2 conjectures predict rational linear relations among Mahler measures of different families when the Jacobian splits or admits elliptic quotients (Meemark et al., 2019, Lalín et al., 2018).

Other formulations appearing in the cited work concern the derived set $T'$ of Salem numbers, greedy $\beta$-expansions for regular Pisot numbers, the number of conjugates outside the unit circle for extremal algebraic integers, and the sets $J_p=\{n\ge 1:\nu_p(H_n)\ge 1\}$ attached to harmonic numbers. These statements are historically unrelated in method, but they share Boyd’s characteristic pattern of experimentally detected arithmetic regularity that later becomes the target of rigorous structural analysis (Amara, 24 Sep 2025, Panju, 2011, Stankov, 2014, Carofiglio et al., 19 Mar 2025).

Context	Typical statement	Status in the cited literature
Mahler measures and elliptic curves	$\beta$0 or rational relations among Mahler measures	Many conductor-specific and family-specific cases proved (Meemark et al., 2019)
Salem and Pisot numbers	$\beta$1	Still described as open in the cited discussion (Amara, 24 Sep 2025)
Regular Pisot $\beta$2-expansions	Co-factors are products of cyclotomic polynomials	Proved for all regular Pisot numbers $\beta$3 approaching $\beta$4 and $\beta$5 (Panju, 2011)
Minimal house / Perron numbers	$\beta$6	Established conditionally for the trinomial model considered (Stankov, 2014)
Harmonic numbers	Finiteness of $\beta$7, density $\beta$8, no $\beta$9	Extensive numerical evidence, not a general proof (Carofiglio et al., 19 Mar 2025)

2. Mahler-measure formulations and elliptic curves

For the Mahler-measure version, the central principle is that a two-variable Laurent polynomial $p$0 defining a genus-one curve should satisfy

$p$1

with $p$2, at least in the tempered elliptic setting. A polynomial $p$3 is tempered when for every side $p$4 of its Newton polygon, the associated edge polynomial $p$5 has only roots of unity as zeros; otherwise it is non-tempered. This distinction is important because the tempered case fits the regulator framework most directly, whereas non-tempered families typically produce additional logarithmic terms (Meemark et al., 2019, Lalín et al., 2015).

Several canonical families recur throughout the literature. One elliptic family is

$p$6

birational to the Deuring model

$p$7

A second family is

$p$8

and a tempered comparison family used in conductor-$p$9 work is

$P$0

For the genus-2 side, important families include

$P$1

$P$2

and

$P$3

depending on the paper and notation. Boyd’s numerical observations linked these families either directly to $P$4 or to each other through rational linear relations reflecting shared elliptic quotients (Lalín et al., 2018, Yang et al., 2022).

The arithmetic explanation is supplied by regulator theory. For a tempered polynomial defining an elliptic curve $P$5, the Mahler measure can be expressed as a regulator integral

$P$6

or equivalently as the value of the elliptic regulator on the $P$7-symbol $P$8. In this form, Boyd’s conjecture becomes a concrete instance of the Beilinson–Bloch paradigm relating regulators to special $P$9-values (Meemark et al., 2019, Lalín et al., 2018).

3. Established elliptic cases: conductors 21 and 30

A decisive class of results concerns explicit conductor-level identities. For the conductor-$$m(P)=\frac{1}{(2\pi)^n}\int_{0}^{2\pi}\cdots\int_{0}^{2\pi}\log\big|P(e^{i\theta_1},\ldots,e^{i\theta_n})\big|\,d\theta_1\cdots d\theta_n$$0 curve attached to

$$m(P)=\frac{1}{(2\pi)^n}\int_{0}^{2\pi}\cdots\int_{0}^{2\pi}\log\big|P(e^{i\theta_1},\ldots,e^{i\theta_n})\big|\,d\theta_1\cdots d\theta_n$$1

the identity

$$m(P)=\frac{1}{(2\pi)^n}\int_{0}^{2\pi}\cdots\int_{0}^{2\pi}\log\big|P(e^{i\theta_1},\ldots,e^{i\theta_n})\big|\,d\theta_1\cdots d\theta_n$$2

was proved by reducing the full Mahler measure to a difference of two half-Mahler measures of the non-tempered curve

$$m(P)=\frac{1}{(2\pi)^n}\int_{0}^{2\pi}\cdots\int_{0}^{2\pi}\log\big|P(e^{i\theta_1},\ldots,e^{i\theta_n})\big|\,d\theta_1\cdots d\theta_n$$3

then evaluating those halves using Ramanujan’s modular-unit parametrization on $$m(P)=\frac{1}{(2\pi)^n}\int_{0}^{2\pi}\cdots\int_{0}^{2\pi}\log\big|P(e^{i\theta_1},\ldots,e^{i\theta_n})\big|\,d\theta_1\cdots d\theta_n$$4 and the Mellit–Brunault regulator formula for modular units (Lalín et al., 2015).

The conductor-$$m(P)=\frac{1}{(2\pi)^n}\int_{0}^{2\pi}\cdots\int_{0}^{2\pi}\log\big|P(e^{i\theta_1},\ldots,e^{i\theta_n})\big|\,d\theta_1\cdots d\theta_n$$5 case was handled by a different but related strategy. The key identity is

$$m(P)=\frac{1}{(2\pi)^n}\int_{0}^{2\pi}\cdots\int_{0}^{2\pi}\log\big|P(e^{i\theta_1},\ldots,e^{i\theta_n})\big|\,d\theta_1\cdots d\theta_n$$6

valid for $$m(P)=\frac{1}{(2\pi)^n}\int_{0}^{2\pi}\cdots\int_{0}^{2\pi}\log\big|P(e^{i\theta_1},\ldots,e^{i\theta_n})\big|\,d\theta_1\cdots d\theta_n$$7 or $$m(P)=\frac{1}{(2\pi)^n}\int_{0}^{2\pi}\cdots\int_{0}^{2\pi}\log\big|P(e^{i\theta_1},\ldots,e^{i\theta_n})\big|\,d\theta_1\cdots d\theta_n$$8 with $$m(P)=\frac{1}{(2\pi)^n}\int_{0}^{2\pi}\cdots\int_{0}^{2\pi}\log\big|P(e^{i\theta_1},\ldots,e^{i\theta_n})\big|\,d\theta_1\cdots d\theta_n$$9. It relates the Mahler measures of a non-tempered family to those of the tempered family $L'(E,0)$0. For generic parameters, the zero loci $L'(E,0)$1 and $L'(E,0)$2 define the same elliptic curve over $L'(E,0)$3 up to isomorphism, with Weierstrass model

$L'(E,0)$4

Using a modular-unit parametrization at level $L'(E,0)$5, the Brunault–Mellit–Zudilin formula, Atkin–Lehner symmetries, and additional functional identities for Mahler measures, the paper proves

$L'(E,0)$6

where all three curves lie in the unique isogeny class of conductor $L'(E,0)$7 (Meemark et al., 2019).

These results are methodologically significant because they show that non-tempered families need not lie outside the regulator-$L'(E,0)$8-value framework. Instead, they often contribute an explicit logarithmic correction. The conductor-$L'(E,0)$9 paper gives representative evaluations such as

$E$0

together with analogous formulas for $E$1. The paper explicitly interprets the logarithmic term as arising from the non-tempered edges of the Newton polygon, as Boyd had conjectured (Meemark et al., 2019).

4. Genus-2 identities, shifted measures, and regulator proofs

Boyd’s conjectures for genus-2 curves predict rational relations between Mahler measures of genus-2 families and those of genus-1 families obtained from elliptic quotients of the Jacobian. One proved instance is

$E$2

together with

$E$3

Lalin and Wu reproved these identities by showing that the relevant $E$4-symbols have identical diamond divisors on a common elliptic curve and that the Mahler-measure cycles correspond to the same periods. This regulator proof complements Bertin–Zudilin’s earlier differential and hypergeometric arguments by making the expected $E$5-value interpretation explicit (Lalín et al., 2018).

A shifted version was later established for Boyd’s genus-2 family

$E$6

The principal identity is

$E$7

where

$E$8

The proof uses elliptic quotients of the genus-2 curve, symbol pushforwards in tame $E$9, diamond-operator computations, and a detailed homology analysis of Deninger paths. In the range $T'$0, this yields explicit formulas such as

$T'$1

$T'$2

thereby extending Boyd’s pattern to a shifted setting (Yang et al., 2022).

A recurrent misconception is that these genus-2 identities are purely formal comparisons of Mahler measures. The regulator literature shows otherwise: the equalities arise because the corresponding measures are realized as regulator integrals on elliptic quotients, and the rational factors are controlled by quotient maps, torsion divisors, and homology multiplicities rather than by ad hoc numerical coincidence (Lalín et al., 2018, Yang et al., 2022).

5. Salem numbers, Pisot numbers, and $T'$3-expansion co-factors

In a different area of number theory, Boyd’s conjecture concerns the set $T'$4 of Salem numbers and its derived set $T'$5. In the formulation quoted by a 2025 manuscript, the conjecture states

$T'$6

where $T'$7 is the set of Pisot numbers. Salem’s classical theorem gives the opposite inclusion $T'$8 in the sense that every Pisot number is an accumulation point of Salem numbers. The 2025 paper claims the stronger statement $T'$9, hence that every accumulation point of Salem numbers belongs to $\beta$0, and concludes that $\beta$1 is closed in $\beta$2 (Amara, 24 Sep 2025).

The same manuscript, however, also states that its own chain of reasoning does not substantiate a disproof of Lehmer’s conjecture and that the broader mathematical status remains that Boyd’s conjecture $\beta$3 and Lehmer’s conjecture remain open. This is an important point of bibliographic interpretation: the article presents a claimed resolution, but its detailed discussion explicitly records a logical gap between the claimed theorem and the stronger conclusion about Mahler measures approaching $\beta$4 (Amara, 24 Sep 2025).

A separate conjecture of Boyd concerns greedy $\beta$5-expansions for regular Pisot numbers below $\beta$6. Boyd had shown that for the regular Pisot numbers approaching $\beta$7 and $\beta$8 with $\beta$9, the co-factor in the factorization of the companion polynomial would always be a product of cyclotomic polynomials, and he conjectured that this holds for all $J_p=\{n\ge 1:\nu_p(H_n)\ge 1\}$0. Panju proved this for all regular Pisot numbers less than $J_p=\{n\ge 1:\nu_p(H_n)\ge 1\}$1 in the $J_p=\{n\ge 1:\nu_p(H_n)\ge 1\}$2- and $J_p=\{n\ge 1:\nu_p(H_n)\ge 1\}$3-families, showing that the true co-factor is cyclotomic after cancellation of cyclotomic factors already present in the defining polynomial. The only non-cyclotomic cases identified in that paper occur in the $J_p=\{n\ge 1:\nu_p(H_n)\ge 1\}$4-based families $J_p=\{n\ge 1:\nu_p(H_n)\ge 1\}$5 with $J_p=\{n\ge 1:\nu_p(H_n)\ge 1\}$6 odd and $J_p=\{n\ge 1:\nu_p(H_n)\ge 1\}$7 with $J_p=\{n\ge 1:\nu_p(H_n)\ge 1\}$8 even, and those co-factors are non-reciprocal, as Boyd predicted (Panju, 2011).

This suggests a useful distinction between two styles of Boyd conjecture. In the Salem/Pisot setting the conjecture is topological and distributional, concerning derived sets in $J_p=\{n\ge 1:\nu_p(H_n)\ge 1\}$9. In the $\beta$00-expansion setting it is algebraic-combinatorial, concerning factorization patterns of Parry polynomials generated by explicit digit expansions (Amara, 24 Sep 2025, Panju, 2011).

6. Minimal house, Perron numbers, and harmonic numbers

Another conjecture attributed to Boyd concerns the extremal distribution of conjugates for algebraic integers of minimal house. If $\beta$01 denotes the minimum of the houses of algebraic integers of degree $\beta$02 that are not roots of unity, and $\beta$03 is the number of conjugates outside the unit circle for an algebraic integer realizing $\beta$04, then Boyd conjectured

$\beta$05

The same paper formulates an analogue for the smallest Perron number of degree $\beta$06. For the trinomials

$\beta$07

it proves that

$\beta$08

At $\beta$09, this gives $\beta$10, and under the Lind–Boyd conjecture for smallest Perron numbers, the paper concludes the Perron analogue of Boyd’s conjecture for that extremal family (Stankov, 2014).

In the arithmetic of harmonic numbers, the relevant objects are

$\beta$11

Eswarathasan–Levine conjectured that $\beta$12 is finite for every prime $\beta$13 and that there are infinitely many harmonic primes, meaning primes for which $\beta$14. Boyd’s probabilistic Galton–Watson model led to three more specific predictions: $\beta$15 should be finite with growth of order $\beta$16, the set of harmonic primes should have density $\beta$17, and very high $\beta$18-adic divisibility should be absent, specifically that there are no pairs $\beta$19 with $\beta$20, and only finitely many if $\beta$21 occurs at all (Carofiglio et al., 19 Mar 2025).

The 2025 computational study verifies the finiteness of $\beta$22 for all primes $\beta$23 with at most one exception, $\beta$24, enumerates harmonic primes up to $\beta$25, finding $\beta$26 harmonic primes among $\beta$27 primes, with proportion approximately $\beta$28, and proves that there are no pairs $\beta$29 with $\beta$30, $\beta$31, for which $\beta$32. These are numerical confirmations rather than a general proof, but they extend Boyd’s earlier computations by factors of about $\beta$33 and $\beta$34, respectively (Carofiglio et al., 19 Mar 2025).

7. Conceptual significance and current status

Across its variants, Boyd’s conjecture is less a single theorem than a programmatic style of arithmetic experimentation. In the Mahler-measure setting, the conjectures have been especially productive because they interact with concrete frameworks: Deninger’s path integral formula, the Bloch regulator, the elliptic dilogarithm, modular units, Atkin–Lehner symmetries, and the Brunault–Mellit–Zudilin formula. This combination has turned many numerical identities into theorems, particularly for conductor $\beta$35, conductor $\beta$36, and several genus-2-to-elliptic reductions (Lalín et al., 2015, Meemark et al., 2019, Lalín et al., 2018, Yang et al., 2022).

The literature also indicates clear limitations. The Brunault–Mellit–Zudilin method applies to elliptic curves admitting modular-unit parametrizations, and the paper on conductor $\beta$37 explicitly notes that such curves are finite in number. In the shifted genus-2 setting, the proof currently covers $\beta$38 and $\beta$39, leaving the intermediate interval untreated by that method. In the $\beta$40-expansion setting, the regular Pisot numbers below $\beta$41 are classified, but irregular Pisot numbers and Pisot numbers $\beta$42 remain open. In the harmonic-number setting, current evidence is extensive but computational. In the Salem-number setting, the cited discussion explicitly treats the broader conjectural status as unresolved despite a claimed stronger theorem in one manuscript (Meemark et al., 2019, Yang et al., 2022, Panju, 2011, Carofiglio et al., 19 Mar 2025, Amara, 24 Sep 2025).

A plausible unifying interpretation is that Boyd’s conjectures identify arithmetic quantities that are experimentally accessible but structurally deeper than their original definitions suggest. In the Mahler-measure cases, the conjectural equality $\beta$43 is now understood as a regulator phenomenon. In the Pisot and Salem cases, accumulation and factorization patterns point toward hidden rigidity in algebraic dynamics. In the harmonic-number case, branching-process heuristics organize $\beta$44-adic divisibility data that would otherwise appear sporadic. The common theme is not a shared formal statement, but the repeated emergence of unexpectedly rigid arithmetic regularity from explicit computation.