Squares with Three Nonzero Digits
نویسنده
چکیده
We determine all integers n such that n2 has at most three base-q digits for q ∈ {2, 3, 4, 5, 8, 16}. More generally, we show that all solutions to equations of the shape Y 2 = t + M · q + N · q, where q is an odd prime, n > m > 0 and t2, |M |, N < q, either arise from “obvious” polynomial families or satisfy m ≤ 3. Our arguments rely upon Padé approximants to the binomial function, considered q-adically.
منابع مشابه
Perfect Powers with Few Ternary Digits
We classify all integer squares (and, more generally, q-th powers for certain values of q) whose ternary expansions contain at most three digits. Our results follow from Padé approximants to the binomial function, considered 3-adically.
متن کاملMinimality and other properties of the width-w nonadjacent form
Let w ≥ 2 be an integer and let Dw be the set of integers which includes zero and the odd integers with absolute value less than 2. Every integer n can be represented as a finite sum of the form n = P ai2 , with ai ∈ Dw, such that of any w consecutive ai’s at most one is nonzero. Such representations are called width-w nonadjacent forms (w-NAFs). When w = 2 these representations use the digits ...
متن کامل#A3 INTEGERS 12A (2012): John Selfridge Memorial Issue PERFECT POWERS WITH FEW TERNARY DIGITS
We classify all integer squares (and, more generally, q-th powers for certain values of q) whose ternary expansions contain at most three digits. Our results follow from Padé approximants to the binomial function, considered 3-adically. –Dedicated to the memory of John Selfridge.
متن کاملDesign and Synthesis of High Speed Low Power Signed Digit Adders
Signed digit (SD) number systems provide the possibility of constant-time addition, where inter-digit carry propagation is eliminated. Such carry-free addition is primarily a three-step process; adding the equally weighted SDs to form the primary sum digits, decomposing the latter to interim sum digits and transfer digits, which commonly belong to {–1, 0, 1}, and finally adding the tra...
متن کاملNew Minimal Weight Representations for Left-to-Right Window Methods
For an integer w ≥ 2, a radix 2 representation is called a width-w nonadjacent form (w-NAF, for short) if each nonzero digit is an odd integer with absolute value less than 2w−1, and of any w consecutive digits, at most one is nonzero. In elliptic curve cryptography, the w-NAF window method is used to efficiently compute nP where n is an integer and P is an elliptic curve point. We introduce a ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2016