bibkey: andrejic2006fibonaccipowers authors: V. Andrejić year: 2006 title: On Fibonacci Powers doi: 10.2298/PETF0617038A url: https://doiserbia.nb.rs/Article.aspx?ID=0353-88930617038A claim: A nonclassical Fibonacci perfect power would imply a Wall–Sun–Sun prime, by prime-index reduction and equality of the entry points at a prime and its square. strata_touched: [] license: citation-only triage: anchor

Perfect powers and the Wall–Sun–Sun obstruction

Verified locator

V. Andrejić, On Fibonacci Powers, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 17 (2006), 38–44, https://doi.org/10.2298/PETF0617038A . Crossref and the DOI Serbia article record identify this publication. The publisher PDF https://doiserbia.nb.rs/ft.aspx?id=0353-88930617038A contains Lemma 4 on page 39 and Theorem 1, its proof and Lemma 5 on page 40.

Claim and scope

Lemma 4 reduces a perfect q-th power F_n with n>12 to a perfect q-th power F_ell at some prime ell>5. Theorem 1 states that the absence of WSS primes would imply that the only Fibonacci perfect powers are 1, 8 and 144. Contrapositively, a further perfect power would yield a WSS prime.

In the proof, every prime factor r of the prime-index block satisfies r^2 dividing F_ell, hence Z(r^2)=Z(r)=ell. Lemma 5 identifies this entry-point equality with the WSS square condition at the signed index r-(r/5).

This supplies a method precedent for FPD.4–FPD.5 of Problems/wall-sun-sun-golden-unit-lift.md. A WSS prime alone does not force all other factors of its rank block to have exponent at least two. The cited implication neither gives its converse nor classifies all powerful Fibonacci values.

OSE. Finite original odd-depth support and a quantitative escape theorem

OSE.0 Fixed object and already-proved prerequisites

This is an ordinary research continuation for the existing owner Problems/wall-sun-sun-golden-unit-lift.md. The deductions below are not attributed to Andrejic’s paper. Retain the ORIGINAL Fibonacci and Lucas sequences, F_0=0, F_1=1, L_0=2, L_1=1. For a prime p>5 put

$$\rho (p)=\min \{n\ge 1:p\mid F_n\},h_p=v_p(F_{\rho (p)})=v_p(F_{p-(5/p)}).$$

WSS means h_p>=2. In this section an odd super-depth prime means h_p is odd and h_p>=3; depths two, four, six, etc. are not in that set. Every use of this condition refers to h_p, not to v_p(F_n) at an arbitrarily multiplied index. Put E_0={1,2,6,12}. Positive one is powerful.

The inspected PERIODIC_TREE.md, ZBD.1 and PBC.1-PBC.4, already proves: (1) powerful Fibonacci values at five-smooth indices occur exactly at E_0; (2) the largest index-prime ell>=7 of a powerful F_n has a powerful F_ell; (3) such a block contains a prime of rank ell and odd h_p>=3; (4) the analogous witnesses at successive largest-prime power layers; (5) the partition of prime ranks into simple-factor and fully exceptional blocks. These are prior repository results, not new OSE theorems. The read snapshot was 9d48e0f0626ad25a583e2453bdda4ad23c70dee9, ZBD/PBC around lines 4180-4375. FPD in the current owner supplies the same largest-prime mechanism. Their square-class input is classical.

At PR head 3dcf7c4b1c1a54cb26bb4ac12787283626ffb28e the parallel local formalization also contains GoldenCubicIndexForm, in addition to the matrix bridge and exact prime-power order source. Its new public endpoint constructs the actual cubic coordinate ring and proves the trace and power-basis determinants. That source was read; it is preserved and not recompiled or claimed as authored here. OSE returns to original prime-depth support, rather than proving another generator-count bound.

We use the following classical inputs with their scopes unchanged:

• If p>5 and rho(p)|n, v_p(F_n)=h_p+v_p(n); otherwise the valuation is zero. Also rho(p)|p-(5/p). The small ranks are rho(2)=3, rho(3)=4, rho(5)=5. See the existing Medina-Rowland note, Theorem 1.4.
• F_n is square at positive indices exactly {1,2,12}. Its positive-index square classes have only two nonsingleton classes, {1,2,12} and {3,6}. Thus a specified squarefree d>2 occurs in F_n=d*y^2 for at most one n. Ribenboim, FFF (2005), (3.4), states this full classification; (3.7) states an effective fixed-d index bound, and (3.8) its rank-closure algorithm. These are stronger inputs than merely the absence of new perfect powers. They are credited rather than reproved here.

OSE.1 A finite, computable closure of a finite prime set

Let S be ANY finite set of primes greater than five. It need not consist of actual exceptional primes. Let H(S) be the least set containing {2,3,5} union S and closed under taking prime divisors of rho(p):

$$H_0=\{2,3,5\}\cup S,H_{r+1}=H_r\cup \bigcup _{p\in H_r}\mathrm{Supp}(\rho (p)),H(S)=\bigcup _{r\ge 0}{H}_r.\text{\hspace{1em}(OSE1)}$$

Here Supp(a) is the prime support of a positive integer, with Supp(1) empty. Write t(S)=|H(S)| and M(S)=product_(p in H(S)) p.

Lemma OSE1. This procedure terminates, and every member of H(S) is at most max({5} union S). For p>5 every prime divisor of rho(p) is STRICTLY smaller than p. H(S) is effectively computable from S using integer factorization and the original recurrence.

Proof. The rank divides one of the even numbers p-1,p+1. An odd prime divisor of either number is at most (p+1)/2<p; the possible prime two is smaller too. At two, three and five, the displayed small ranks add only primes already in {2,3,5}. Thus no new prime exceeds the initial maximum, and a strictly increasing chain of these finite subsets must terminate. At termination it is closed, and induction shows it is contained in every closed set containing H_0. Ranks can be computed by iterating the invertible Fibonacci pair recurrence modulo p, or by removing prime factors from p-(5/p) and testing actual Fibonacci zeros. The same rank-closure operation occurs in Ribenboim (3.8). Its use below is to control original odd-depth support of powerful values.

OSE.2 The original odd-depth support controls both index and square class

For every n>=1 define the finite set of external odd-depth divisors

$$U(n)=\{p>5\text{ prime}:p\mid F_n,p\nmid n,h_p\text{ odd}\}.$$

The word external means only the explicit condition p not dividing n. Let d(n)=product_(p|F_n, v_p(F_n) odd) p be the positive squarefree kernel; F_n=d(n)*y^2 for a positive integer y.

Theorem OSE2. If U(n) is contained in S, then

$$\boxed{\mathrm{Supp}(n)\subseteq H(S),d(n)\mid M(S).}\text{\hspace{1em}(OSE2)}$$

Proof. Suppose n has a prime divisor outside H(S), and choose the largest such divisor ell. It is at least seven. Every prime p|F_ell has rho(p)=ell and p>ell, by the prior prime-index argument FPD3/ZBD1. If p divided n, maximality of ell among the OUTSIDE primes would give p in H(S). Closure would then imply ell in H(S), a contradiction. Thus every prime p|F_ell is prime to n. The valuation formula gives v_p(F_n)=h_p=v_p(F_ell), since ell divides n.

The nonsquare F_ell has some prime divisor p with odd valuation. For this p the preceding equalities show p in U(n), hence p in S and H(S). Closure again implies ell=rho(p) in H(S), the required contradiction. This proves the first inclusion. Notice that ell need not be the largest prime of n; it is the largest one OUTSIDE the specified closed set. This is the additional descent step not supplied by merely taking P+(n).

If q divides d(n) but q is outside H(S), then q>5 and, by the first inclusion, q does not divide n. Since q|F_n, its odd value exponent is h_q+v_q(n)=h_q. Therefore q in U(n) subset S subset H(S), another contradiction. The kernel is squarefree, so its support inclusion is exactly the asserted divisibility. No power-basis index is used.

OSE.3 A cardinality bound for all powerful values supported by S

For n>=1 put

$$T(n)=\{p>5\text{ prime}:p\mid F_n,h_p\ge 3\text{ odd}\},$$

$$\mathcal{P}(S)=\{n\ge 1:F_n\text{ powerful and }T(n)\subseteq S\}.$$

T(n) may include primes dividing n; this only strengthens its containing U(n) in the powerful case. S may be an arbitrary finite prime set, so membership in P(S) does not assume a classification of all WSS primes.

Theorem OSE3. For every finite S as above,

$$\boxed{n\in \mathcal{P}(S)⟹\mathrm{Supp}(n)\subseteq H(S),d(n)\mid M(S),}\text{\hspace{1em}(OSE3)}$$

$$\boxed{\#\mathcal{P}(S)\le 2^{t(S)}-4.}\text{\hspace{1em}(OSE4)}$$

In particular P(S) is finite, without abc, a density premise, or an assumption about the number of depth-two WSS primes.

Proof. If p in U(n) and F_n is powerful, then p does not divide n, so v_p(F_n)=h_p is odd and at least two. Hence h_p>=3 and p in T(n). Thus U(n) subset T(n) subset S, and OSE2 proves OSE3.

There are exactly 2^t squarefree divisors of M(S). Eight divide 30, since {2,3,5} subset H(S). A powerful F_n with d(n)|30 must have a five-smooth index: otherwise take its largest index-prime ell>=7. A prime divisor p of odd exponent in F_ell exists by nonsquareness; p>ell and p does not divide n, so its same odd exponent in F_n puts p>5 in d(n), a contradiction. The prior five-smooth classification therefore forces n in E_0. Conversely these four indices are powerful and have T(n) empty, so all four belong to P(S).

For every remaining squarefree divisor d of M(S), d does not divide 30 and in particular d>2. The classical full square-class theorem gives at most one positive n with d(n)=d, before even imposing powerfulness or T(n) subset S. Thus there are at most 2^t-8 further indices. Adding the four classical ones proves OSE4. The collisions {1,2,12} and {3,6} are handled by the explicit E_0 count; they are not silently treated as singleton square classes.

Finite-escape corollary. Among more than 2^t(S)-4 distinct powerful Fibonacci INDICES, at least one value has a prime p outside S with odd original h_p>=3. In particular infinitely many powerful Fibonacci values would force infinitely many such odd-depth super-WSS primes.

Proof. Otherwise all those indices would be in P(S), contradicting OSE4. If the global odd-super-depth set were finite, take it for S. The number of indices is then bounded by OSE4. Distinct positive values and indices differ only at F_1=F_2, so infinitude is equivalent.

Conditional consequence, with the hypothesis displayed. If S_odd={p>5:h_p>=3 odd} is finite, then the TOTAL set of powerful Fibonacci indices has size at most 2^|H(S_odd)|-4. This permits any number of WSS primes with even depth, including h=2 or h=4. It does not prove that S_odd is finite or that any WSS prime exists. When S_odd is empty the classification reduces to E_0, the already-known ZBD/FPD consequence.

OSE.4 Effective finite reduction and a fixed family of elliptic curves

For a specified finite S, OSE3 reduces ALL candidates for P(S) to

$$F_n=d{y}^2,d\mid M(S),d\text{ squarefree},n,y\ge 1.$$

Ribenboim’s cited (3.7) gives an effectively computable N(d) for this fixed-d equation. Consequently

$$N_S=\underset{d\mid M(S)}{\max }N(d)$$

is a valid finite search bound. Factoring the finitely many F_n up to N_S and checking the original h_p computes P(S). The existence of these classical effective bounds is used; no numerical N_S or complete search is reported in this continuation. A scan with a chosen cutoff is not a substitute for N_S. The kernel uniqueness bound OSE4 does not itself provide a height bound.

There is also an exact geometric realization of the candidate equations. Put epsilon=(-1)^n. The ORIGINAL golden Pell relation gives L_n^2=5d^2 y^4+4epsilon, so

$$\boxed{(X,Y)=(5d^2{y}^2,5d^{2}y{L}_n)\in E_{d,ϵ}:Y^2=X^3+20ϵd^{2}X.}\text{\hspace{1em}(OSE5)}$$

The discriminant is -64*(20epsilon*d^2)^3, nonzero for d>0. The map follows from the exact polynomial identity

$$(5d^{2}yz{)}^2-(5d^2{y}^2{)}^3-20ϵd^2(5d^2{y}^2)=25d^4{y}^2(z^2-5d^2{y}^4-4ϵ).$$

There are only 2^(t+1) specified curves. For recovering P(S), retain X/(5d)=F_n, X/(5d^2)=y^2 with y positive, Y/(5d^2y)=L_n positive, epsilon=(-1)^n, powerfulness, and the original-depth support test. Arbitrary rational or integral points without these filters are not WSS witnesses. Siegel’s classical theorem independently gives finiteness of the integral points on each curve, but the sharper OSE4 uses the full square-class uniqueness theorem and no numerical rank calculation.

OSE.5 Two disjoint exact-rank channels and their odd-depth budget

For a prime ell>=7 keep the two ACTUAL integers F_ell and L_ell. Their prime supports are disjoint. For p|F_ell and q|L_ell respectively,

$$\rho (p)=\ell ,\pi (p)=4\ell ,p\equiv 1(\mathrm{mod}4),$$

$$\rho (q)=\pi (q)=2\ell ,(5/q)=1,$$

and the factor exponents are their original h_p,h_q. Supports for all different ell, including between the two channels, are disjoint.

Proof. Exact ranks and prime-to-index valuations are the prior FPD/PH block results. At an odd zero rank ell, the Fibonacci matrix Q^ell is scalar aI with a^2=-1 modulo p. Hence its scalar order is four, p=1 modulo four, and its pair period is 4ell. For q|L_ell, the norm identity L_ell^2-5F_ell^2=-4 gives (5/q)=1, and the matrix trace-zero identity gives Q^(2ell)=I; its zero rank is already 2ell. Finally the products F_ell*L_ell=F_(2ell) are pairwise coprime for distinct prime ell, because gcd(F_(2ell),F_(2ell’))=F_2=1.

The classical Fibonacci and Lucas square classifications ensure an odd-exponent prime in EACH channel. Each witness therefore has either h=1 or odd h>=3. Let N_1(X) and N_o(X) count rational primes p<=X with rho(p) equal to ell or 2ell for some prime ell>=7, and respectively h_p=1 or odd h_p>=3. Let Pi(Y) count ordinary rational primes <=Y and phi=(1+sqrt(5))/2. For real Y>=7,

$$\boxed{N_1({\varphi }^Y)+N_o({\varphi }^Y)\ge 2(\mathrm{Pi}(Y)-3).}\text{\hspace{1em}(OSE6)}$$

Proof. Choose an odd-exponent factor from each of the two channels at every prime ell<=Y. All chosen primes are distinct and smaller than phi^ell, by the original Binet formulas. Their original depths put each in exactly one of the two displayed counts. There are Pi(Y)-3 indices. No comparison prime outside the actual Fibonacci/Lucas supports is used.

Thus, by the classical prime number theorem,

$$\underset{X\to \infty }{\mathrm{liminf}}\frac{(N_1(X)+N_o(X))\log \log X}{\log X}\ge \frac{2}{\log \varphi }.\text{\hspace{1em}(OSE7)}$$

If N_o(X)=o(log(X)/log(log(X))), the same lower bound holds for N_1 alone. This is a CONDITIONAL non-WSS conclusion. Applied separately, each channel has constant 1/log(phi): one locates pair period 4ell and p=1 modulo four, the other split primes with period 2ell. The older PBC.4 already proves the Fibonacci-channel partition and a conditional order-of-growth bound; those statements are credited, not recounted. OSE6 records both disjoint channels, their original odd-depth alternative, and the explicit exponential scale together.

For a residue refinement requiring only the elementary mod-eight and mod-four recurrences: if ell=5 or7 modulo12, F_ell=5 modulo8, so one odd-depth factor p is five modulo eight (all its factors are one modulo four). If ell=5 or11 modulo12, L_ell=3 modulo4, so one odd-depth factor q is three modulo four. Neither observation distinguishes depth one from odd depth at least three by itself.

OSE.6 Formalization targets and remaining arithmetic obligation

The local proof work is kept separate from the user’s concurrent Lean files. The proposed theorem order for later formalization is:

1. Finite rank closure: strict descent above the fixed small-prime set, stabilization, leastness and support closure.
2. Largest-prime-OUTSIDE-closure descent OSE2, using the actual Fibonacci rank, valuation and prime-index nonsquare inputs.
3. Powerful U(n) subset T(n), squarefree-kernel support, then the finite set injection proving the exact bound OSE4 with its eight base classes.
4. OSE5 as a coordinate polynomial identity plus explicit arithmetic filters. For OSE6 use finite disjoint supports before any asymptotics.

The full square-class theorem is a genuine classical prerequisite for OSE4, stronger than a determinant identity or the prime-power order lemma. It has been used in prior repository written theory; no claim that it is already Lean-certified follows. Without that prerequisite, OSE2/OSE3’s support conclusions and the explicit curve reduction remain, and Siegel still supplies qualitative finiteness. No formal axiom or new Lean wrapper is inserted in this pass.

The independent WSS question still requires deciding an original h_p or excluding an actual unbounded exceptional family. OSE does not prove N_o is small, construct a nonclassical powerful Fibonacci value, supply a WSS prime, or exclude the unrestricted P^2 Q^3 ternary-block pattern. It gives a finite-support obstruction and a cardinality theorem on the original odd-super-depth set. Finiteness of that set remains a displayed hypothesis only where explicitly stated.

OSE.7 Primary literature and scope of the verification

• P. Ribenboim, FFF: (Favorite Fibonacci Flowers), Fibonacci Quarterly 43(1) (2005), 3-14. Sections (3.4)-(3.5), printed page 8, give the square classes; (3.7)-(3.8), page 9, give fixed-square-class effectiveness and the earlier rank-closure algorithm. The square-class result is attributed there to Square classes of Fibonacci and Lucas numbers, Portugaliae Mathematica 46(2) (1989), 159-175. https://www.fq.math.ca/Papers1/43-1/paper43-1-1.pdf The parsed passages were read. Web screenshot requests for pages 8-9 failed, as did the container download, so no visual-page check is claimed.
• Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Annals of Mathematics 163 (2006), 969-1018, DOI 10.4007/annals.2006.163.969. https://annals.math.princeton.edu/2006/163-3/p05 The publisher abstract corroborates the square exceptions. It is not substituted for the stronger fixed-square-class uniqueness theorem.
• N. Fellini and M. Ram Murty, Wieferich primes in number fields and the conjectures of Ankeny-Artin-Chowla and Mordell, arXiv:2508.08472v2, Theorems 1.3-1.4 and 4.3: https://arxiv.org/html/2508.08472v2 The paper already proves non-Wieferich infinitude and a log X/log log X lower bound under finiteness of ALL base-alpha super-Wieferich primes, and recalls Siegel’s theorem. OSE does not claim that general mechanism as new. Its hypothesis selects original odd Fibonacci depths and its finite-support cardinality statement concerns powerful Fibonacci indices. No abc hypothesis, random quotient model or unrestricted Chebotarev prime is inserted into the displayed proofs.

The rank-closure operation, square-class theorem, primitive-prime facts, valuation theory and fixed-d effectiveness are classical. ZBD/PBC’s odd-super witness and Fibonacci-channel count were already in the repository. The additional finite-support descent and explicit counting bound are proved here with those prerequisites, without claiming global priority or an externally stated open problem solved.

verify_ose.py is an exact finite diagnostic, not a proof of a finite super-depth hypothesis. It checks actual F_n for 1<=n<=60, rank closures, original versus multiplier depth, and both prescribed prime-index channels through ell=43. Integer factorizations are proposed by SymPy and every factor is independently certified by trial division. It also checks the elliptic map as a polynomial identity and the exceptional square classes in the tested range. No new WSS sample or complete integral-point solver is reported. No Lean/Scribe compilation or independent-model review is claimed; the unrestricted arguments are the ordinary proofs above.