The Density of Primes Dividing a Term in the Somos-5 Sequence
نویسندگان
چکیده
The Somos-5 sequence is defined by a0 = a1 = a2 = a3 = a4 = 1 and am = am−1am−4+am−2am−3 am−5 for m ≥ 5. We relate the arithmetic of the Somos-5 sequence to the elliptic curve E : y2 + xy = x3 + x2 − 2x and use properties of Galois representations attached to E to prove the density of primes p dividing some term in the Somos-5 sequence is equal to 5087 10752 .
منابع مشابه
Elliptic curves and related sequences
A Somos 4 sequence is a sequence (hn) of rational numbers defined by the quadratic recursion hm+2 hm−2 = λ1 hm+1 hm−1 + λ2 h 2 m for all m ∈ Z for some rational constants λ1, λ2. Elliptic divisibility sequences or EDSs are an important special case where λ1 = h 2 2, λ2 = −h1 h3, the hn are integers and hn divides hm whenever n divides m. Somos (4) is the particular Somos 4 sequence whose coeffi...
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