Parametric Algorithms for the 5-Modular Analog of ES (Sierpiński): Structure of Solutions, Parameterization, and Constructive Proofs (SERP)

Parametric Algorithms for the 5-Modular Analog of ES (Sierpi´nski): Structure of Solutions, Parameterization, and Constructive Proofs (SERP) E. Dyachenko dyachenk[email protected] November 25, 2025 Abstract We consider the problem of representing the fraction 5 P as a sum of three distinct unit fractions: 5 P=1 A+1 B+1 C, A < B < C, A, B, C ∈N. We analyze the case of primes P≡ 1 ( mod 5), for which two types of solutions are distinguished: ED1 (exactly one denominator divisible by P , namely C = cP ) and ED2 (exactly two denominators divisible by P , namely B = bP and C = cP ). Parametric constructions and enumeration algorithms are developed, including transitions between types of solutions. The paper proposes a deterministic algorithm based on searching for the intersection of a parametric lattice, defined by a pair ( αi, d′ i ), with the box Bk ( T ). For any fixed prime P≡ 1 ( mod 5) the algorithm constructively produces a solution. Using analytic methods (Bombieri–Vinogradov theorem, Chebotarev theorem), it is shown that the density of admissible parameters is high, which ensures polylogarithmic search complexity in the average case, i.e., for most primes. A strict complexity guarantee for all primes remains conditional and depends on the finite covering hypothesis. This study continues the work for coefficient 4(the Erd˝os–Straus conjecture) [ 2 ]; here a similar structure of parameterization and solutions is extended to coefficient 5. In the analytic applications, averaging tools are provided, used for density estimates in parametric boxes. 1 Introduction The Sierpi´nski problem in the 5-modular variant is formulated as follows: for a prime P one must find natural numbers A<B<Csatisfying 5 P=1 A+1 B+1 C.(1.1) 2020 Mathematics Subject Classification: Primary 11N05, 11P21; Secondary 94A60, 20P05. Key words and phrases: factorization, Bombieri–Vinogradov large sieve, the larger sieve of Greaves, deterministic algorithms; analytic number theory; number theory. * Licence: Text is available under the Creative Commons NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) 1 Analogous to the Erd˝os–Straus conjecture for 4 P , here the coefficient 5leads to a change in the structure of parameterizations and to new types of solutions. For primes P≡ 1 ( mod 5) two constructive classes are distinguished: •ED1: exactly one denominator divisible by P(without loss of generality C=cP ); •ED2: exactly two denominators divisible by P(namely B=bP and C=cP). 1.1 Transition from 4/P to 5/P The methods of parameterization and proofs from [ 2 ] are preserved in the present study, but the core formulas change in the transition 4→5. In particular: •estimate for the minimal denominator: P < 5A < 3P; •in ED1 the relation takes the form 5c−1 = γP (instead of 4c−1 = γP); •in ED2 the core takes the form t= 5bc −b−cand (5b−1)(5c−1) = 5Pδ + 1. Thus, the problem 5 /P is a direct analogue of the case 4 /P , but requires new constructive techniques. Unlike factorization-based approaches used in classical works on ESC, here methods of finite covering of residue classes and lattice algorithms are applied. These tools allow one to prove the existence of solutions for every prime P≡ 1 ( mod 5) and ensure polylogarithmic complexity of the search algorithm. 2 Motivation The parameterization of the equation and the division into multiplicity configurations ( P|B and/or P|C ) make it possible to construct explicit search procedures and to prove the existence of solutions for the case P≡ 1 ( mod 5). This approach not only guarantees the existence of solutions but also defines the structure of families in arithmetic progressions. The possibility of transferring the constructive method from coefficient 4to coefficient 5 was correctly noted in [ 7 ]. In the present work this transfer is carried out explicitly within the framework of the ED1/ED2 methods developed in [ 2 ]. The key element here is the mechanism of finite covering of residue classes, which guarantees the existence of solutions for every prime P≡ 1 ( mod 5) without recourse to factorization. Thus, the motivation of the study is to show that constructive methods based on lattices and covering of residue classes allow one to move from asymptotic reasoning to effective algorithms with polylogarithmic complexity. 3 Ordering For the classes P≡ 4 , 3 , 2 ( mod 5) explicit decompositions are available; the remaining case is P≡1 (mod 5). 3.1 Explicit decompositions for P≡ 1 (mod 5) Let P= 5P′+ 4 be prime: 5 P=5 5P′+ 4 =1 P′+ 1 +1 2·(P′+ 1) ·(5P′+ 4) +1 2·(P′+ 1) ·(5P′+ 4). Let P= 5P′+ 3 be prime: 5 P=1 P′+ 1 +1 (P′+ 1)(5P′+ 3) +1 (P′+ 1)(5P′+ 3). 2 Let P= 5P′+ 2 be prime (here P′is odd): 5 P=1 P′+ 1+2 (P′+ 1) ·(5P′+ 2)+1 (P′+ 1) ·(5P′+ 2) =1 P′+ 1+1 (P′+1) 2·(5P′+ 2)+1 (P′+ 1) ·(5P′+ 2), where the third summand is further detailed as a sum of two parts, since P′ + 1 is even. The case P= 5P′+ 1 is the subject of this paper. 3.2 Estimate for the minimal denominator If A≤B≤Cand A, B, C ∈N, then 5 P>1 A⇒5A > P ⇒P < 5A. Moreover, from A≤B≤Cit follows that 5 P=1 A+1 B+1 C≤3 A⇒5A≤3P, and, excluding the degenerate case A=B=C, we obtain strict bounds: P < 5A < 3P. (3.1) 3 4 Notation Main parameters Pprime number >2, with 5∤P; often P≡1 (mod 5),P≡2 (mod 5),P≡3 (mod 5),P≡4 (mod 5) P′integer: P= 5P′+ 1 A, B, C denominators in the Sierpi´nski hypothesis formula, ordered A≤B≤C H(S) Sierpi´nski hypothesis: 5 P=1 A+1 B+1 C Parameters of the ED1 and ED2 methods γ5c−1 P(or 5b−1 Pin the second case); often γ≡4 (mod 5),gcd(γ, c)=1or gcd(γ, b) = 1 u, v “multipliers” in the identity: for ED1 u=γA −c,v=γB −c,uv =c2; for the variant with b:u=γA −b,v=γC −b,uv =b2; always u≤v. The factorization dimension in the case of 5 is achieved by passing to normalized parameters uand v. δ t =P·δ, with ED2 requiring δ|bc b, c in ED2 B=bP and/or C=cP t t = 5bc −b−c r, s r = 5b−1,s= 5c−1, with r≡s≡4 (mod 5),rs = 5Pδ + 1 ggcd(b, c) b′, c′decomposition b=b′g,c=c′g(normalized form) d′square factor in the decomposition of δ αsquarefree factor in the decomposition of δ Transitions between methods CED1(P)set of admissible quadruples (γ, c, u, v)for ED1 CED2(P)set of admissible triples (δ, b, c)for ED2 yminimal divisor of 5c−1, with y≡3 (mod 5) P′′ modulus for ED1 in convolution, defined via γby the formula ... ED2→ED1 transition: A=bc δ,B=bP;u=γA −c,v=γB −c ED1→ED2 transition: A=u+c γ,b=v+c γP ,δ=bc A Anticonvolution algorithm for the reverse step ED1→ED2 according to the formulas above Lattices and boxes kdimension of the vector parameter u0(P)vector shift for the affine class Λ,Λjsublattices of Zkof index Mor Mj M, Mjindices of sublattices Bk(T)box {u∈Zk: 1 ≤ui≤T} B(I) P,B(II) Pboxes of type I/II with additional conditions GP(T)admissible parameters in the box G∗ Pclass of admissible quadruples (γ, c, u, v)for ED1, satisfying: γ∈N,c∈Z,u=γA −c,v=γB −c,uv =c2,u≤v Analytic notation a PLegendre symbol; for composite modulus the a γ(Jacobi symbol, for prime P) is used π(y)prime counting function ≪,≫,≍standard asymptotic symbols Dset δ≤X,δ≡4 (mod 5) T(δ)prime counting function in progression, see Appendix §A 4 5 Parametrization of ED1 (one multiple, C=cP) 5.1 Kernel and identities Let C=cP. From (1.1) it follows that (5c−1)AB =cP(A+B).(5.1) Setting γ=5c−1 P∈N, we obtain (γA −c)(γB −c) = c2.(5.2) Hence γ≡4 (mod 5) and gcd(γ, c)=1(since 5c≡1 (mod γ)). Lemma 5.1. We always have gcd(γ, c)=1. Proof. From 5c−1 = γP we obtain 5c≡1 (mod γ), hence gcd(γ, c)=1. 5.2 Full parametrization of ED1 with filters by P Theorem 5.2. Let P≡ 1 ( mod 5), γ≡ 4 ( mod 5),5 c− 1 = γP , gcd ( γ, c ) = 1. Let u, v ∈N such that uv =c2, u ≡v≡ −c(mod γ), u ≡ −c(mod P), v ≡ −c(mod P). Then A=u+c γ, B =v+c γ, C =cP give a solution of (1.1) of type ED1 with P∤A, B . Conversely, every ED1 -solution generates such γ, c, u, v. Proof. From (5.1) it follows that (5.2) holds. Setting u = γA −c , v = γB −c , we obtain uv = c2 and u≡v≡ −c ( mod γ ), whence γ| ( u + c ), γ| ( v + c )and A, B ∈N . The conditions u≡ −c ( mod P ), v≡ −c ( mod P )are equivalent to P∤A , P∤B thanks to gcd ( γ, c )=1and 5c−1 = γP. Reversibility follows from (5.2). 5.3 Multiplicity filters by P From A= (u+c)/γ,B= (v+c)/γ,5c−1 = γP we have P|A⇐⇒ u≡ −c(mod P), P |B⇐⇒ v≡ −c(mod P). For ED1 it is required simultaneously that u≡ −c(mod P)and v≡ −c(mod P). 5.4 Example: P= 11 The minimal γ≡ 4 ( mod 5) with 5 c− 1 = γP gives γ = 4, c = (4 · 11 + 1) / 5=9. Divisors of c2 = 81, compatible with u≡ −c≡ 3 ( mod 4) and u≡ −c≡ 2 ( mod 11), include u = 3, v = 27. Then A=3+9 4= 3, B =27 + 9 4= 9, C = 99,1 3+1 9+1 99 =5 11. 5 6 Parametrization of ED2 (two multiples, B=bP,C=cP) 6.1 Setup and identities Consider 5 P=1 A+1 bP +1 cP , A < bP ≤cP, P ∤A. Multiplying by AbcP , we obtain A(5bc −b−c) = Pbc. (6.1) Let t:= 5bc −b−c=Pδ,δ∈N. Then A=bc δ.(6.2) Equivalently, (5b−1)(5c−1) = 5Pδ + 1.(6.3) Setting r= 5b−1,s= 5c−1, we obtain rs = 5Pδ + 1 and r≡s≡4 (mod 5). 6.2 Full parametrization of ED2 Theorem 6.1. Let Pbe prime and δ∈N. Let r, s ∈Nsuch that r s = 5Pδ + 1, r ≡s≡4 (mod 5). Set b= (r+ 1)/5,c= (s+ 1)/5. If δ|bc, then A=bc δ, B =bP, C =cP give a solution of (1.1) of type ED2 . Under permutation r↔s we have b↔c and B↔C ; ordering B≤Cis achieved by choosing r≤s. Moreover, for b≤cwe have A≤B. Proof. From (6.1) and t = Pδ it follows that (6.2) holds. Equality (6.3) is obtained from (5 b− 1)(5 c− 1) = 25 bc − 5 b− 5 c + 1 = 5(5 bc −b−c ) + 1 = 5 Pδ + 1. The congruences r≡s≡ 4 (mod 5) are obvious. For b≤cwe have A≤B⇐⇒ bc δ≤bP ⇐⇒ c≤Pδ = 5bc −b−c⇐⇒ c(5b−2) ≥b, which holds for b≥1,c≥b. 6.3 Example: P= 11 Take δ = 1. Then 5 Pδ + 1 = 56 = 4 · 14,4 ≡ 14 ≡ 4 ( mod 5). The pair r = 4, s = 14 gives b = 1, c= 3,A= 3,B= 11,C= 33. Verification: 1 3+1 11 +1 33 =11 + 3 + 1 33 =15 33 =5 11. 6 7 Multiplicity configurations with respect to P and classification Lemma 7.1. In any solution of (1.1)at least one of A, B, C is divisible by P. Proof. Multiplying (1.1) by ABCP :5 ABC = P ( AB + AC + BC ). Modulo P :5 ABC ≡ 0, hence P|ABC. Lemma 7.2. It is impossible that A=aP,B=bP ,C=cP simultaneously. Proof. Then 5 = 1/a + 1/b + 1/c ≤3— impossible. Lemma 7.3. The minimal denominator Ais not divisible by P. Proof. From (3.1) we have 5A < 3P; if A=aP, then 5A≥5P > 3P, a contradiction. Proposition 7.4. Every solution of (1.1) for prime P = 5 falls into exactly one of the classes: - ED1 : exactly one denominator divisible by P (without loss of generality C = cP ), with P∤A, B ; - ED2 : exactly two denominators divisible by P (namely B = bP , C = cP ), with P∤A . The cases “none” and “all three” are excluded by Lemmas 7.1 and 7.2, and Lemma 7.3 excludes P|A . 7.1 Intermediate transition to parameters b′, c′ From Proposition 7.4 it follows that every solution of (1.1) for P≡ 1 ( mod 5) belongs either to class ED1 or to class ED2. In both cases it is convenient to isolate normalized parameters that describe the part of denominators divisible by P. • In the case ED1 we have C = cP , and the parameters γ, c arise from normalizing the condition 5c−1 = γP. • In the case ED2 we have B = bP , C = cP , and the parameters b′, c′ arise from the kernel (5b−1)(5c−1) = 5Pδ + 1. Thus, in the case ED2 the problem reduces to analyzing pairs ( b′, c′ )with additional divisibility and congruence conditions. 7.2 Parametric box in coordinates (b′, c′) For a fixed threshold Tconsider the set Bb′,c′(T) = {(b′, c′)∈N2: 1 ≤b′, c′≤T, b′< c′}. This set contains all candidates for solutions in the original parameters. Additional conditions (e.g., gcd ( b′, c′ )=1, congruences modulo d , divisibility 4 b′c′ = P + d ) are imposed as filters on the points (b′, c′). 7.3 Box threshold via Aand P From the inequality P < 5A < 3P it follows that the minimal denominator A always lies in the interval ( P/ 5 , 3 P/ 5). This provides natural bounds for the parametric box. Definition 7.5. For a fixed prime Pwe define the box in the original parameters (b′, c′)as Bb′,c′(P) = {(b′, c′)∈N2: 1 ≤b′, c′≤3P/5, b′< c′}. 7 7.4 Transition to coordinates (x, y) Under the linear transformation x=b′+c′, y =c′−b′, the image of the set Bb′,c′(P)is the sublattice Bx,y(P) = {(x, y)∈Z2:x≡y(mod 2), x > y > 0, x, y ≤6P/5}. 7.5 Condition for d′ In the original system the parameter d′ appears as the square factor in the decomposition of δ . In the new coordinates the condition d′|(b′+c′)is rewritten as d′|x. Proposition 7.6. The existence of a point ( x, y ) ∈Bx,y ( P )satisfying the conditions x≡y ( mod 2), y > 0, x, y ≤ 6 P/ 5and d′|x , is equivalent to the existence of an admissible pair ( b′, c′ ) and hence to a solution of (1.1). 7.6 Corollary Defining the box Bb′,c′ ( T )and transferring it to coordinates ( x, y ), we obtain a lattice with simple linear conditions: - parity x≡y(mod 2), - order y > 0, - bound x, y ≤2T, - divisibility d′|x. Thus, the existence of a solution is equivalent to the presence of a point ( x, y )in the box Bx,y ( T )satisfying these conditions. This makes the proof constructive and allows the use of methods of finite covering of residue classes. Example Table 1 Table 1: ED2-decompositions for P= 73 P A B C b c δ α, d′ 73 15 584 8760 8 120 64 α= 1, d′= 8 73 15 657 3285 9 45 27 α= 3, d′= 3 73 15 730 2190 10 30 20 α= 5, d′= 2 73 15 876 1460 12 20 16 α= 1, d′= 4 7.7 Geometry of the surface and thickenings Consider the quadratic surface F(δ, b, c) = (5b−1)(5c−1) −5Pδ −1 = 0. The thickening of this surface, given by the condition |F| ≤ ∆in the parameter space ( δ, b, c ), yields the set of ED2 candidates. Since we work in the context of discrete values, it is critical to estimate the number of solutions satisfying modular conditions. Lemma 7.7 (Window in δ ).For fixed values of b and c , and for ∆ ≥ 0, the number of integers δsatisfying the condition |F(δ, b, c)| ≤ ∆does not exceed: 1 + 2∆ 5P. Proposition 7.8 (Estimate of thickening size).In the box where b∈ [ B, 2 B ]and c∈ [ C, 2 C ], the total number of triples (δ, b, c)for which the condition |F| ≤ ∆holds can be estimated as: ≪1 + ∆ PBC +B+C. Remark 7.9.For pairs ( b, c )subject to the condition δ|bc , the number of such pairs in the corresponding rectangle is ≪BC δτ(δ) + B+C. 8 8 Constructive geometry of ED2 for the Sierpi´nski hypothesis (preserving the logic of ESC) 8.1 Setup and kernel, fully analogous to ESC For a prime P≡1 (mod 5) we prove the existence of a solution 5 P=1 A+1 B+1 C, B =bP, C =cP, P ∤A, b =c. Multiplying by AbcP and introducing the parameter δ∈N, we obtain A(5bc −b−c) = P bc, 5bc −b−c=Pδ, A =bc δ. Quadratic kernel ED2: (5b−1)(5c−1) = 5Pδ + 1. This is fully isomorphic to the kernel for ESC ( k = 4) under the substitution 4 7→ 5: all checks and constructions transfer verbatim. 8.2 Normalization and linear system (as in ESC) Let b=g b′,c=g c′, where g=αd′and gcd(b′, c′)=1, and δ=α(d′)2, α squarefree. Then the canonical conditions (ED2) take the linear form b′c′=M=Aα, b′+c′=m d′, m = 5A−P > 0. In coordinates x=b′+c′,y=c′−b′we have x=m d′,x2−y2 4=Aα, x ≡y(mod 2). This is exactly the same affine lattice of finite index as in the ESC section, with the coefficient k replaced. 8.3 Parametric box and geometric covering (transfer from ESC) The minimal denominator is bounded by P 5<A<3P 5. The projection of the lattice Λ ED2 onto the ( x, y )-plane lies in a box of linear size O ( P ); the diagonal period equals d′ , and for H, W ≥d′ the intersection of the box with the lattice is nonempty. The lattice index does not depend on P , as in ESC, which stabilizes the density of admissible points and ensures constructive covering without factorization. 8.4 Back-test filters and non-degeneration (identical to ESC scheme) For any assembled row we check: (5b−1) ≡(5c−1) ≡4 (mod 5), δ |bc, gcd(b′, c′) = 1, b′+c′=m d′, b′c′=Aα, A =bc δ∈Z,P 5<A<3P 5. Order and distinctness: b = c , consistent ordering of denominators (according to your editorial scheme), without degeneration. 9 Comment. Summation of the main term follows from Proposition B.1 and linearity; summation of error terms gives the stated remainder. The estimate of P 1 /φ (5 r )is standard: φ (5 r ) = 4 φ ( r ) when gcd(r, 5) = 1, and Pn≤R, (n,m)=1 1/φ(n) = c(m) log R+O(1). Proposition B.5 (Exceptional set via the larger sieve).Let R≤x1/2/ ( log x ) C . Then the number of r≤R,r≡4 (mod 5),gcd(r, 5δ)=1, for which the progression P≡1 (mod 5), P ≡ −(5δ)−1(mod r) contains no primes P≤x, satisfies ≪Rexp −clog R log log R, where c > 0is an absolute constant (follows from Theorem A.2). Remark B.6 (Chebotarev: additional local filters).If necessary, one can simultaneously impose splitting conditions for r and/or s = (5 Pδ + 1) /r in a fixed extension of number fields; by Theorem A.3 the corresponding classes have positive natural density, and intersection with the specified AP retains positive density. How to use in the algorithm. - Preselection of r : fix several small r≡ 4 ( mod 5) (or enumerate r≤R in the BV range). - Pre-sieving by progressions: for each r precompute the class P≡ − (5 δ ) −1 ( mod r )and combine with P≡ 1 ( mod 5) (CRT). - Enumeration of P : scan primes P in the combined classes; by Proposition B.1 the expected frequency is ≍ 1 /φ (5 r ). - Verification of ED2 : for a found P compute s = (5 Pδ + 1) /r , check s≡ 4 ( mod 5), and reconstruct b = ( r + 1) / 5, c = ( s + 1) / 5, A = bc/δ . - Balancing: choosing R≤x1/2/ ( log x ) C gives an optimal compromise between the number of progressions and control of errors (BV), while Proposition B.5 guarantees the smallness of the exceptional set of r. Double summation: average over primes P Define for fixed δand R≥2the quantity N(P;R, δ) = #nr≤R:r≡4 (mod 5),gcd(r, 5δ)=1, r |(5Pδ + 1) o. This is the number of “local” parameters r for a given prime P . Then for R≤x1/2/ ( log x ) C we have the average asymptotic 1 #{P≤x:P≡1 (mod 5)}X P≤x P≡1 (5) N(P;R, δ) = C5,δ log R+O(1),(B.1) where C5,δ >0is a constant depending only on δand classes modulo 5. Idea of proof. Change the order of summation: X P≤x P≡1 (5) N(P;R, δ) = X r≤R r≡4 (5) gcd(r,5δ)=1 #{P≤x:P≡1 (5), P ≡ −(5δ)−1(r)}. For each rapply Proposition B.1 (modulus 5r) and sum the main terms: X r Li(x) φ(5r)= Li(x)X r≤R r≡4 (5) gcd(r,5δ)=1 1 φ(5r)= Li(x)C5,δ log R+O(1), and the total error is controlled linearly in r using Proposition B.1. Division by the number of primes P≤x,P≡1 (5), gives (B.1). 16 Corollary B.7 (Average supply of local parameters).For R = ( log x ) B with fixed B > 0, the average value of N ( P ; R, δ )over primes P≤x , P≡ 1 ( mod 5), grows as C5,δ Blog log x + O (1). In particular, the average number of admissible rtends to infinity. Remark B.8 (On estimates of “most P ”).The transition from the average to a statement of the form “for most P there exists at least one r≤R ” can be obtained by second moment methods or via the larger sieve (Proposition B.5) with a coordinated choice of R and x . These details are not required for the constructive part of the algorithm, but they explain its good average behavior. 17