Asymptotic Fermat for signatures ( r, r, p ) using the modular approach

Let K be a totally real ﬁeld, and r ≥ 5 a ﬁxed rational prime. In this paper, we use the modular method as presented in the work of Freitas and Siksek to study non-trivial, primitive solutions ( x, y, z ) ∈ O 3 K of the signature ( r, r, p ) equation x r + y r = z p (where p is a prime that varies). An adaptation of the modular method is needed, and we follow the work of Freitas which constructs Frey curves over totally real subﬁelds of K ( ζ r ). When K = Q we get that for most of the primes r < 150 with r (cid:4)≡ 1 mod 8 there are no non-trivial, primitive integer solutions ( x, y, z ) with 2 | z for signatures ( r, r, p ) when p is suﬃciently large. Similar results hold for quadratic ﬁelds, for example when K = Q ( √ 2) there are no non-trivial, primitive solutions ( x, y, z ) ∈ O 3 K with √ 2 | z for signatures (5 , 5 , p ), (11 , 11 , p ) , (13 , 13 , p ) and suﬃciently large p .


Introduction
The modular method for solving Diophantine equations was pioneered by Frey, Serre, Ribet, and Wiles with the famous proof of Fermat's Last Theorem in 1995 [31], [28]. Since then, various generalizations of Fermat's Last Theorem have been considered, that are of the shape Ax p + By q = Cz r with 1/p + 1/q + 1/r < 1 (1) for fixed integers A, B and C. We call (p, q, r) the signature of the equation (1). A primitive solution (x, y, z) is a solution where x, y and z are pairwise coprime and a non-trivial solution (x, y, z) is a solution where xyz = 0. The so-called Generalized Fermat equation (1) is the subject to the following conjecture, which is known to be a consequence of the ABC-conjecture.
Conjecture 1. Fix A, B, C ∈ Z pairwise coprime. Then, there exist only finitely many non-trivial, primitive triples with (x, y, z) ∈ Z 3 and p, q, r primes such that (1) holds (here solutions where one of x, y, z equals 1 are counted only once, e.g. 2 3 + 1 q = 3 2 for all q counts as one solution.) A partial result towards this conjecture due to Darmon-Granville [10] asserts the following.
Theorem 2 (Darmon-Granville). For A, B, C fixed as above and a fixed signature (p, q, r) there exists only finitely many non-trivial, primitive integer solutions to (1).
Furthermore, Conjecture 1 has been established for many families of signatures (both over Q and over totally real number fields) using variants of the modular approach. Some examples of signatures that have been (partially) solved using the modular method can be found in Table 1.
The modular method is a strategy of attacking Diophantine Equations which can be summarised in three steps.
Step 1. Constructing a Frey elliptic curve. Attach to a putative solution (of some Diophantine equation) an elliptic curve E/K, for K an appropriately chosen totally real number field. We require E to have the Artin conductor N p bounded independently of the putative solution.
Step 2. Modularity/Level lowering. Prove modularity of E/K and irreducibility of some residual Galois representationsρ E,p attached to E, to conclude (via level lowering results), thatρ E,p corresponds to a (Hilbert) newform.
Step 3. Contradiction. Prove that among the finitely many (Hilbert) newforms predicted above, none of them corresponds toρ E,p .
Let us focus on signature (r, r, p). Usually, the equation considered is where r, p are rational primes, d is a positive integer, with r and d fixed. When employing the modular method, we find that the first step requires more work.
In this paper, we will follow the recipes described by Freitas in [14] to construct the desired Frey curve. Freitas describes a framework for attacking (2), by constructing the so-called multi-Frey family of curves. However, we will only work with one elliptic curve belonging to this family, as described in Section 2.
Another popular direction that originated in Darmon's program involves using Frey abelian varieties of higher dimensions (in place of Frey curves). In a recent paper, Billerey, Chen, Dieulfait, Freitas, and Najman [5] use some of the additional structure of the Frey abelian varieties to get asymptotic resolutions for (11, 11, p) for solutions locally away from xy = 0. In order to accomplish this, the authors had to first build on the progress surrounding the modular method (Step 2).
An even more recent result by Freitas and Najman [16] gives the following.
Theorem 3. Let d = 1 be a positive integer that is divisible only by primes q ≡ 1 mod r. Then, for a set of primes p with positive density, the equation (2) has no non-trivial, primitive integer solutions (x, y, z) with 2|(x + y) or r|(x + y).
Our work aims to complement their resolution by focusing on d = 1. In this context, we say the asymptotic Fermat Last Theorem holds for signature (r, r, p) if there exists a constant B r such that, there are no non-trivial, primitive integer solutions (respecting certain local conditions) to the equation x r + y r = z p for p > B r .
Moreover, we examine the equation (2) over a totally real number field K, and give computable criteria of testing if the asymptotic Fermat Last Theorem for signature (r, r, p) with non-trivial, primitive solutions in O K (respecting certain local conditions) holds. The argument is pioneered by Freitas and Siksek in [18] involving modularity, an "Eichler-Shimura"-type result, image of inertia comparison, and the study of S-unit equations.

Our Results
Let K be a number field. For any rational prime p we denote by S K,p := {P : P is a prime of K above p}.
Moreover, for a fixed prime r, we denote by K + r := K(ζ r + ζ −1 r ), where ζ r is a primitive r th root of unity. If the prime r is understood from the context, we will omit the subscript r in K + r . We would like to study the solutions of x r + y r = z p over a totally real field K via the modular approach. The first step is to construct a Frey elliptic curve E. One way to realize this is by defining E/K + r as described in section 2. Therefore, it makes sense to view our equation over K + r first. Note that if K is totally real, then K + r stays totally real. Theorem 4 (Main Theorem). Let r ≥ 5 be a rational prime and K a totally real number field. Define K + := K(ζ r + ζ −1 r ) and S K + := S K + ,2 ∪ S K + ,r . Suppose that there exists some distinguished prime P ∈ S K + ,2 , such that every solution satisfies max(|v P (λ)|, |v P (µ)|) ≤ 4v P (2). Then, there is a constant B K,r (depending only on K) such that for each rational prime p > B K,r , the equation x r + y r = z p has no non-trivial, primitive solutions (x, y, z) ∈ O 3 K + with P|z. In particular, by considering Q to be the prime in S K,2 below P, the equation x r +y r = z p with p > B K,r has no non-trivial, primitive solutions (x, y, z) ∈ O 3 K with Q|z.

Remark 5.
• If we assume modularity of elliptic curves over totally real number fields, the constant B K,r is effectively computable.
• By Siegel [27], S-unit equations have finitely many solutions over number fields. Moreover, they are effectively computable, for example, an S-unit solver has been implemented in the free open-source mathematics software, Sage by A. Alvarado, A. Koutsianas, B. Malmskog, C. Rasmussen, D. Roe, C. Vincent, M. West in [1].
Then, there is a constant B := B K,r (depending only on r and K) such that for each rational prime p > B, the equation x r + y r = z p has no non-trivial, primitive solutions (x, y, z) ∈ O K + with P|z.
Corollary 7. Fix r ≥ 5 a rational prime such that r ≡ 1 mod 8. Let Q + := Q(ζ r +ζ −1 r ), suppose that 2 is inert in Q + and 2 ∤ h + Q + . Then, there is a constant B r (depending only on r) such that for each rational prime p > B r , the equation x r + y r = z p has no non-trivial, primitive, integer solutions with 2|z.  [30][pg. 412] to see that for the above values of r the associated relative class number h − r is odd, so indeed 2 ∤ h + Q + . The rest of the conditions in the hypothesis can be easily checked (for example by using MAGMA).
Let's now study x r + y r = z p over quadratic fields of the form K( √ d) for d square-free.
Corollary 9. Fix r ≥ 5 a rational prime and let K := Q( √ d) with d a squarefree, positive integer and K + := K(ζ r + ζ −1 r ). Assume that (i) r ∤ d and r ≡ 1, d mod 8; (ii) there is a unique prime P above 2 in K + and we denote the unique prime of K below it by Q; (iii) 2 ∤ h + K + . Then, there is a constant B K,r (depending only on r and K) such that for each rational prime p > B K,r , the equation x r + y r = z p has no non-trivial, primitive solutions (x, y, z) ∈ O 3 K with Q|z. Example 10. One can check the conditions in the hypothesis (for example by using MAGMA) to get the following.

Notational conventions
Let K be a totally real field and E/K be an elliptic curve of conductor N E . Let p be a rational prime. Define the following quantities: where ∆ q is the minimal discriminant of a local minimal model for E at q. Note that in literature (for example in [14]), N p is commonly referred to as the Artin conductor of E outside p and it is denoted by N (ρ E,p ). Let G K = Gal(K/K). For an elliptic curve E/K, we write for the representation of G K on the p-torsion of E.
For a Hilbert eigenform f over K, we let Q f denote the field generated by its eigenvalues. A comprehensive definition of Hilbert modular forms and their associated representation can be found, for example in Wiles' [32].
If S is a finite set of primes of K we denote by Moreover, O * S will be its unit group. An ideal I of O K is called a prime-to-S-ideal if its prime decomposition contains no primes in S.
Acknowledgements. I am sincerely grateful to my supervisor Samir Siksek for his continuous support, useful discussions, and for reviewing this paper.

Constructing Frey Elliptic Curves
2.1 Diophantine Equations Related to x r + y r = z p Let r ≥ 5 be a fixed rational prime and K a totally real number field. Consider the equation viewed over K + := K(ζ r + ζ −1 r ) which, as we noted, is totally real. In this section, we follow Freitas' ideas in [14, Section 2] to relate the primitive solutions of (4) to primitive solutions of several homogeneous Diophantine equations defined over K + . We write Over the field L := K(ζ r ) one gets the factorization Proposition 11. Suppose that (x, y, z) is a coprime, non-trivial solution of (4) in O 3 L . Then, any two factors x + ζ i r y and x + ζ j r y with 0 ≤ i < j ≤ r − 1 are coprime outside S L,r .
Proof. Let p / ∈ S L,r be a prime of L. Suppose by a contradiction that p|(x+ζ i r y) and p|(x + ζ j r y) for 0 ≤ i < j ≤ r − 1. Then is an S L,r -unit and p / ∈ S L,r , it follows that p|y. This and the fact that p|(x + ζ i r y), implies that p|x contradicting the fact that x and y are coprime.
Since r ≥ 5 is a prime, r − 1 ≥ 4 is even and φ r factors over K + into degree two factors of the form Moreover, we consider f 0 (x, y) = (x + y) 2 .
Then the factors f i (x, y) and f j (x, y) are coprime outside S K + ,r for 0 ≤ i < j ≤ r−1 2 and each can be factorized as follows Proof. Firstly we note that for 0 ≤ k ≤ r−1 2 , so by Lemma 11 f k (x, y) are coprime outside S L,r . Thus, they are coprime outside S K + r . Moreover, Putting these together, and the fact that f 0 (x, y) = (x + y) 2 we get the conclusion.
We will now define a Frey elliptic curve by generalizing to a totally number field the recipes described in [14].
Lemma 13. The numbers A, B, C factorize in O K + is as follows where Z A , Z B , Z C are coprime prime-to-S K + ,r -ideals dividing (z)O K + , P|Z A with exponents a pr , b pr , c pr ≥ 0 as given in Proposition 12.
Proof. First note that α, β, γ can be written in the form ±ζ r (1 − ζ t r )(1 − ζ u r ), where neither t nor u are multiples of r, which means that the only primes dividing αβγ are the ones in S K + ,r . The result then follows from the definition of A, B, C and Proposition 12.
Note 14. We note that the Frey elliptic curve E depends on the prime p as the coefficients A, B and C do.

Arithmetic Invariants
It is a standard result that an elliptic curve E/K defined as with A + B + C = 0 has the corresponding arithmetic invariants Proposition 15. Let (x, y, z) be a primitive non-trivial solution to (4). Then, the conductor of E is Then, for all primes p / ∈ S K + , the model E is minimal, semistable and satisfies p|v p (∆ E ). Moreover where 0 ≤ e ′ P ≤ e P and 0 ≤ f ′ q ≤ f q and N E , N p are as in (3). Proof. It is enough to show that for p / ∈ S K + the following holds (i) if p ∤ ABC, the model E is minimal and has good reduction at p; (ii) if p|ABC the model E is minimal and has multiplicative reduction at p and moreover p|v p (∆ E ).
Then, the result will follow from the definitions of N E and N p given in (3). In order to prove (i) and (ii) we note that a prime p / If p ∤ ABC, then v p (∆ E ) = 0. Thus, the model is minimal and E has good reduction at p. If p|ABC, then p divides precisely one of A, B and C as they are coprime outside S K + ,r ⊂ S K + . Hence, v p (c 4 ) = 0, giving a minimal model for E with multiplicative reduction at p. By (14) we see that p|v p (∆ E ).

Modularity
Let K be a totally real field and E an elliptic curve over K, we say that E is modular if there exists a Hilbert cuspidal eigenform f over K of parallel weight 2, with rational Hecke eigenvalues, such that the Hasse-Weil L-function of E is equal to the Hecke L-function of f. In particular, this implies that the mod p Galois representations are isomorphic, which we denote by ρ E,p ∼ ρ f,p . We will use the following modularity theorem proved by Freitas, Hung, and Siksek in [15]: Theorem 16. Let K be a totally real field. There are at most finitely manyKisomorphism classes of non-modular elliptic curves E over K. Moreover, if K is real quadratic, then all elliptic curves over K are modular.
Furthermore, Derickx, Najman, and Siksek have recently proved in [12]: Theorem 17. Let K be a totally real cubic number field and E be an elliptic curve over K. Then E is modular.
Corollary 18. Let E p be the Frey curve defined by (11) (which has a dependency on p as noted in 14). By construction, E p is defined over a totally real field which we denote by K. Then, there is some constant A K depending only on K, such that E p is modular whenever p > A K .
Proof. By Theorem 16, there are at most finitely many possibleK-isomorphism classes of elliptic curves over K which are not modular. Let E be the elliptic curve defined in (11). Let j 1 , j 2 , . . . , j n ∈ K be the j-invariants of these classes.
Each equation j(λ) = j i has at most six solutions λ ∈ K. Thus there are values λ 1 , . . . , λ m ∈ K (where m ≤ 6n) such that if λ = λ k for all k, then the elliptic curve E with j-invariant j(λ) is modular.
If λ = λ k then −B/A = λ k . Hence, as ideals By (12) we get that where Z is a prime-to-S K + ,r -ideal, with P|Z and integer exponents l pr := b pr − a pr . Then by (15) and (16), v P (λ k ) = pv P (Z) > 0. Thus p|v P (λ k ). As λ k is fixed, it gives a lower bound on p for each k, and by taking the maximum of these bounds we get A K .
Remark 19. The constant A K is ineffective as the finiteness of Theorem 16 relies on Falting's Theorem (which is ineffective). See [15] for more details. Note that if K is quadratic or cubic we get A K = 0 (by the last part of Theorem 16 and Theorem 17).

Irreducibility of mod p representations of elliptic curves
We need the following theorem in the level lowering step of our proof. This was proved in [17, Theorem 2] and it is derived from the work of David and Momose who in turn built on Merel's Uniform Boundedness Theorem.
Theorem 20. Let K be a Galois totally real field. There is an effective constant C K , depending only on K, such that the following holds. If p > C K is prime, and E is an elliptic curve over K which has multiplicative reduction at all q|p, then ρ E,p is irreducible.
Remark 21. The above theorem is also true for any totally real field by replacing K by its Galois closure.

Level Lowering
We present a level lowering result proved by Freitas and Siksek in [18] derived from the work of Fujira [22], Jarvis [21], and Rajaei [25].
Theorem 22. Let K be a totally real field and E/K be an elliptic curve of conductor N E , suppose the following statements hold: (i) p ≥ 5, the ramification index e(q/p) < p − 1 for all q|p, and Q(ζ p ) + K, Then, there is a Hilbert eigenform f of parallel weight 2 that is new at level N p (as defined in 3) and some prime ̟ of Q f such that ̟|p and ρ E,p ∼ ρ f,̟ .

Image of Inertia
We gather information about the images of inertia ρ E,p (I q ). This is a crucial step in controlling the behaviour at the primes in S K + of the newform obtained by level lowering.
Lemma 23. Let E be an elliptic curve over K with j-invariant j E . Let p ≥ 5 and let q ∤ p be a prime of K. Then p|#ρ E,p (I q ) if and only if E has potentially multiplicative reduction at q (i.e. v q (j E ) < 0) and p ∤ v q (j E ).
For the rest of this section let (x, y, z) ∈ O 3 K + be a non-trivial, primitive solution to (4) and P ∈ S K + ,2 with P|z. We define the Frey curve E as in (11).
Proof. Firstly, note that α, β, γ ∈ O * S K + ,r , i.e. their factorisation contains only primes above r. Moreover, A, B, C are coprime outside S K + ,r (by Proposition 11 and Proposition 12). By the definition of A, we see that P|A, hence P ∤ B, C. Thus, by (13) v P (j E ) = 8v P (2) − 2v P (A). We note that v P (A) = v P (f k1 (x, y)) = pv P (z), hence completing the proof.
Lemma 25. Let E, (x, y, z) and P as above and the prime exponent p > 4v P (2).Then E has potentially multiplicative reduction at P and p|#ρ E,p (I P ).
Proof. Assume that P ∈ S K + ,2 with v P (z) = k. By Lemma 24 and the fact that p > 4v P (2), it follows that v P (j) < 0 and clearly p ∤ v P (j i ). This implies that E has potentially multiplicative reduction at P and by Lemma 23 we get p|#ρ E,p (I P ).

Eichler-Shimura
For totally real fields, modularity reads as follows.
Conjecture 26 (Eichler-Shimura). Let K be a totally real field. Let f be a Hilbert newform of level N and parallel weight 2, with rational eigenvalues. Then there is an elliptic curve E f /K with conductor N having the same Lfunction as f.
Theorem 27. Let E be an elliptic curve over a totally real field K, and p be an odd prime. Suppose that ρ E,p is irreducible, and ρ E,p ∼ ρ f,̟ for some Hilbert newform f over K of level N and parallel weight 2 which satisfies Q f = Q. Let q ∤ p be a prime ideal of O K such that: (i) E has potentially multiplicative reduction at q, Then there is an elliptic curve E f /K of conductor N with the same L-function as f.

Proof of Theorem 4
Firstly, we put together the first steps of the modularity approach to get the following.
Theorem 28 (Level Lowering and Eichler Shimura). Fix r ≥ 5 a rational prime. Let K be a totally real number field and define K + := K(ζ r + ζ −1 r ). Suppose there exists a distinguished prime P ∈ S K + ,2 . Then there is a constant B K,r depending only on K such that the following hold. Suppose (x, y, z) ∈ O 3 K + is a non-trivial, primitive solution to x r +y r = z p with prime exponent p > B K,r such that P|z. Write E for the Frey curve (11). Then, there exists an elliptic curve E ′ over K + such that (i) the elliptic curve E ′ has good reduction outside S K + ; (iv) E ′ has potentially multiplicative reduction at P (v P (j E ) < 0) .
Proof. The proof follows precisely as the one of Theorem 9 in [18]. We will include it here for completion.
We first observe by Proposition 15 that E has multiplicative reduction outside S K + . From Corollary 16 it follows that E is modular and by Theorem 20 that ρ E,p is irreducible. Applying Theorem 22 and Proposition 15 we see that ρ E,p ∼ ρ f,̟ for a Hilbert newform f of level N p and some prime ̟|p of Q f . Next we reduce to the case when Q f = Q, after possibly enlarging B K,r . This step uses standard ideas originally due to Mazur that can be found in [2, Section 4], [8,Proposition 15.4.2], and so we omit the details.
Next, we want to show that there is some elliptic curve E ′ /K + of conductor N p having the same L-function as f. We apply Lemma 25 and by possibly enlarging B K,r get that E has potentially multiplicative reduction at P and p|#ρ E,p (I P ). The existence of E ′ follows from Theorem 27 after possibly enlarging B K,r to ensure that p ∤ (Norm K + /Q (P) ± 1). By putting all the pieces together we can conclude that there is an elliptic curve E ′ /K of conductor N p satisfying ρ E,p ∼ ρ E ′ ,p . This proves (i) and (ii).
To prove (iii) we use that ρ E,p ∼ ρ E ′ ,p for some E ′ /K + with conductor N p . After enlarging B K,r by an effective amount, and possibly replacing E ′ by an isogenous curve, we may assume that E ′ has full 2-torsion over K + . This uses standard ideas which can be found, for example, in [26,. Now let j E ′ be the j-invariant of E ′ . As we have already seen, Lemma 25 implies p|#ρ E,p (IP), hence p|#ρ E ′ ,p (IP), thus by Lemma 23 we get that E ′ has potentially multiplicative reduction atP and so vP(j E ′ ) < 0. Now we are ready to prove the Theorem 4.
Proof of Theorem 4. The proof follows precisely as the one of Theorem 3 in [18]. The idea consists in associating to each non-trivial, primitive solution (x, y, z) ∈ O 3 K + with P|z a Frey elliptic curve as described in Theorem 28, which we denote by E. Let p > B K,r as given in Theorem 28, then E gives rise to an elliptic curve The condition on the S K + -units in our hypothesis gives v P (j ′ ) ≥ 0, a contradiction. This is proven by Freitas and Siksek in [18,Theorem 3] by parametrizing all of the curves E ′ with the above properties. In particular, the last line follows by the fact that any non-trivial, coprime solution (x, y, z) ∈ O 3 K with Q|z gives rise to a solution (x, y, z) ∈ O 3 K + with P|z.

Proof of Theorem 6
Suppose that K is a totally real number field and r ≥ 5 is a fixed rational prime. By (i) r is inert in K. Denote K + := K(ζ r + ζ −1 r ) and let Note that π r is the uniformizer of the unique prime above r in K + . By (ii) there is a unique prime P ∈ S K + ,2 of ramification index e := e(P/2) and by (iii) 2 ∤ h + K + . Moreover, we have that the congruence π r ≡ ν 2 mod P (4e+1) has no solutions in ν ∈ O K + /P (4e+1) by (iv).
We will show that these assumptions guarantee that any solution to an S K + -unit equation satisfies the hypothesis in Theorem 4, giving the desired conclusion.
Lemma 29. Fix r ≥ 5 and K a totally real number field satisfying (i)-(iv) as described above. Consider the natural homomorphism Proof. Let µ ∈ Ker(ϕ), so µ ≡ 1 mod P 4e+1 . We want to show that µ is a square. Consider π r as described in (17), a uniformizer for the unique prime above r in K + . Thus, µ ∈ O * S K + ,r , implies that µ = ǫπ t r , where ǫ ∈ O * K + . We first show that ǫ must be a square. Note that ). As P is the unique prime above 2, we get that (4)O K + = P 2e and hence π r ≡ α 2 mod P 2e .

This belongs to O K + [X]
and has discriminant ∆ = ǫ ∈ O * K + , proving that L is unramified at all the finite places, contradicting 2 ∤ h + K + . Thus, we must have ǫ := γ 2 for some γ ∈ O * K + . Putting everything together µ = ǫπ t r = γ 2 π t r . In order to show that µ is a square, it is enough to show that t is even. So, let's suppose that t is odd, i.e. t = 2k + 1 for some integer k. Hence µ = (γπ k r ) 2 π r . As we assumed µ ≡ 1 mod P 4e+1 , it must be that case that π r ≡ ν 2 mod P 4e+1 for some ν ∈ O K + , contradicting assumption (iv). In conclusion t must be even and so µ ∈ (O * Proof of Theorem 7. This is an application of Theorem 4. We first note that by using the notation at the beginning of this section we can consider P r := (π r )O K + to be the unique prime above r in K + and P to be the unique prime above 2 in K + where we denote by e := e(P/2) the ramification index of P. Hence S K + ,r = {P r } and S K + ,2 = {P} giving S K + = {P, P r }.
Suppose by a contradiction we have an S K + -unit solution (λ, µ) with |v P (λ)| > 4e. Without loss of generality, we may assume v P (λ) > 4e. Otherwise, one can consider ( 1 λ , − µ λ ) instead, which is also a solution to the equation. By the properties of non-archimedean valuations applied to (18) it follows that v P (µ) = 0. Thus, we deduce that µ ≡ 1 mod P 4e+1 . Hence µ lies in the kernel of the natural homomorphism By Lemma 29, Kerϕ ⊆ (O * S K + ,r ) 2 . Thus, for each solution of the S K + -unit equation (18) with v P (λ) > 4e, v P (µ) = 0 (19) we get that µ = τ 2 with τ ∈ O * S K + ,r . As there are only finitely many solutions to the S-unit equation, we may suppose that (λ, µ) satisfies (19) with the value of v P (λ) as large as possible.
We can rewrite (18) as Denote by λ 1 := 1 − τ and λ 2 := 1 + τ and by t i := v P (λ i ) for i = 1, 2. By assumption t := v P (λ) > 4e, giving t = t 1 + t 2 > 4e. By noting that we can only have that t 1 = e or t 2 = e. By changing the sign of τ if necessary, we may assume that t 2 = e and hence t 1 = t − e. Now, note that by rearranging we get the following S K + -unit relation: We compute v P ( to (18) with valuations v P (λ ′ ) > t, v P (µ) = 0. This contradicts the maximality of v P (λ) and completes the proof.

Proof of Corollaries 7 and 9
Lemma 30. Let P be the only prime above 2 in K + and λ ∈ O K + . Suppose that λ ≡ ν 2 mod P n where ν ∈ O K + . Then where Q is the unique prime above 2 in K and e ′ is the ramification index e ′ := e(P/Q) and v ∈ O K .
Proof. Taking norms of λ ≡ ν 2 mod P n we get that As λ, ν ∈ O K + , it follows that Norm K + /K (λ), Norm K + /K (ν) ∈ O K and so Now, we fix a prime r ≥ 5 and K a totally real number field. We let K + := K(ζ r + ζ −1 r ). Then, as in the statement of Theorem 6 we assume r is inert in K, and P is the unique prime above 2 in K + with ramification index e := e(P/2) and we denote by π r := ζ r + ζ −1 r − 2.
Proof of Corollary 9. Using the above notation with K := Q( √ d), we will show that the conditions on (i) and (ii) imply that π r ≡ ν 2 mod P 4e+1 with ν ∈ O K + and then the result follows from Theorem 6. Note that as r ∤ d it follows that r is inert in K. Note that as there is a unique prime above 2 in K we have the following two cases. 2 ]. In this case 2 is inert in K giving e = e ′ = e(P/2). Suppose by a contradiction that π r ≡ ν 2 mod P 4e+1 with ν ∈ O K . By Lemma 30 it follows that r ≡ v 2 mod 2 5 ≡ r mod 32 The first equation implies that b is even. If a would be even too, it would imply that r is even which is a contradiction. Note that this implies that a 2 ≡ 1 mod 8 and b 2 ≡ 0, 4 mod 8. Thus, using the second equation we get either r ≡ 1 mod 8 or r ≡ d mod 8, contradicting the hypothesis. In this case, 2 is totally ramified in K and we denote by Q the unique prime above 2 in K. Hence Q 2 = (2)O K and e := e(P/2) = 2e(P/Q) = 2e ′ . Suppose by a contradiction that π r ≡ ν 2 mod P 4e+1 with ν ∈ O K . By Lemma 30 it follows (in particular) that r ≡ v 2 mod Q 4e/e ′ , giving r ≡ v 2 mod 16 for As The first equation implies that 8|ab. As before, a and b cannot be simultaneously even, so we have two cases.
Case 1. If 8|b (and a is odd) then the second equation gives a 2 ≡ r mod 16 contradicting the assumption that r ≡ 1 mod 8.
Case 2. If 8|a (and b is odd) then the second equation gives b 2 d ≡ r mod 16. In particular, b 2 d ≡ r mod 8 and the only odd square modulo 8 is 1, thus it implies d ≡ r mod 8 contradicting the assumption that r ≡ d mod 8.
Hence, both (i) and (ii) imply that π r ≡ ν 2 mod P 4e+1 with ν ∈ O K + , so we can conclude the proof by Theorem 6.