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Let I denote the set of all irrational numbers, θ ∈ I, and simple continued fraction expansion of θ be [a 0 , a 1 , . . . , an, . . .]. Then a 0 is an integer and {an} n≥1 is an infinite sequence of positive integers. Let Mn(θ) = [0, an, a n−1 , . . . , a 1 ] + [a n+1 , a n+2 , . . .]. Then the set of numbers {lim sup Mn(θ) | θ ∈ I} is called the Lagrange Spectrum L. Notably 3 is the first cluster point of L. Essentially lim inf L or lim L = 3. Perron [Über die approximation irrationaler Zahlen durch rationale, I, S.-B. Heidelberg Akad. Wiss., Abh. 4 (1921) 17 pp;Über die approximation irrationaler Zahlen durch rationale, II, S.-B. Heidelberg Akad. Wiss., Abh. 8 (1921) 12 pp.] has found that lim inf{lim sup Mn(θ) | θ = [a 0 , a 1 , a 2 , . . . , an, . . .] and an ≥ 3 frequently} = (65+9 √ 3) 22 . This article forwards the value of lim inf{lim sup Mn(θ) | θ = [a 0 , a 1 , . . . , an, . . .] and an ≥ 4 frequently}, a long awaited cluster point of Lagrange Spectrum.

In this article, we have proposed a generalized Lucas matrix (recursive matrix of higher order) having relation with generalized Fibonacci sequences and established many special properties in addition to that usual matrix algebra. Further, we have proposed a modified public key cryptography using these matrices as keys in Affine cipher and key agreement for encryption-decryption with the combination of terms of generalized Lucas sequences under residue operations. In this scheme, instead of exchanging the whole key matrix, only a pair of numbers(parameters) need to be exchanged, which reduces the time complexity as well as space complexity of the key transmission and has a large key-space.

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