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A word is square-free if it does not contain any square (a word of the form XX), and is extremal square-free if it cannot be extended to a new square-free word by inserting a single letter at any position. Grytczuk, Kordulewski, and Niewiadomski proved that there exist infinitely many ternary extremal square-free words. We establish that there are no extremal square-free words over any alphabet of size at least 15.Recently, Grytczuk, Kordulewski, and Niewiadomski [5] introduced the study of extremal squarefree words.Definition 1.1. An extension of a finite word W is a word W ′ = W 1 xW 2 , where x is a single letter and W 1 , W 2 are (possibly empty) words such that W = W 1 W 2 . An extremal square-free word W is a square-free word such that none of its extensions is square-free.The only binary extremal square-free words are 010 and 101. Via a delicate construction, Grytczuk et al. showed in [5] that there exist infinitely many ternary extremal square-free words. They and Pawlik also raised several open problems concerning larger alphabet sizes in [5] and [6], including the nonexistence of extremal square-free words over an alphabet of size 4. Mol and Rampersad [7] then classified all possible lengths of extremal ternary square-free words. Conjecture 1.2 ([5], [7]). There exists no extremal square-free word over a finite alphabet of size at least 4.

Inspired by Lehmer’s conjecture on the non-vanishing of the Ramanujan $$\tau $$ τ -function, one may ask whether an odd integer $$\alpha $$ α can be equal to $$\tau (n)$$ τ ( n ) or any coefficient of a newform f(z). Balakrishnan, Craig, Ono and Tsai used the theory of Lucas sequences and Diophantine analysis to characterize non-admissible values of newforms of even weight $$k\ge 4$$ k ≥ 4 . We use these methods for weight 2 and 3 newforms and apply our results to L-functions of modular elliptic curves and certain K3 surfaces with Picard number $$\ge 19$$ ≥ 19 . In particular, for the complete list of weight 3 newforms $$f_\lambda (z)=\sum a_\lambda (n)q^n$$ f λ ( z ) = ∑ a λ ( n ) q n that are $$\eta $$ η -products, and for $$N_\lambda $$ N λ the conductor of some elliptic curve $$E_\lambda $$ E λ , we show that if $$|a_\lambda (n)|<100$$ | a λ ( n ) | < 100 is odd with $$n>1$$ n > 1 and $$(n,2N_\lambda )=1$$ ( n , 2 N λ ) = 1 , then $$\begin{aligned} a_\lambda (n) \in&\{-5,9,\pm 11,25, \pm 41, \pm 43, -45,\pm 47,49, \pm 53,55, \pm 59, \pm 61,\\&\pm 67, -69,\pm 71,\pm 73,75, \pm 79,\pm 81, \pm 83, \pm 89,\pm 93 \pm 97, 99\}. \end{aligned}$$ a λ ( n ) ∈ { - 5 , 9 , ± 11 , 25 , ± 41 , ± 43 , - 45 , ± 47 , 49 , ± 53 , 55 , ± 59 , ± 61 , ± 67 , - 69 , ± 71 , ± 73 , 75 , ± 79 , ± 81 , ± 83 , ± 89 , ± 93 ± 97 , 99 } . Assuming the Generalized Riemann Hypothesis, we can rule out a few more possibilities leaving $$\begin{aligned} a_\lambda (n) \in \{-5,9,\pm 11,25,-45,49,55,-69,75,\pm 81,\pm 93, 99\}. \end{aligned}$$ a λ ( n ) ∈ { - 5 , 9 , ± 11 , 25 , - 45 , 49 , 55 , - 69 , 75 , ± 81 , ± 93 , 99 } .

In this paper, we exhibit an irreducible Markov chain M on the edge k-colorings of bipartite graphs based on certain properties of the solution space. We show that diameter of this Markov chain grows linearly with the number of edges in the graph. We also prove a polynomial upper bound on the inverse of acceptance ratio of the Metropolis-Hastings algorithm when the algorithm is applied on M with the uniform distribution of all possible edge k-colorings of G. A special case of our results is the solution space of the possible completions of Latin rectangles.