Proof Without Words: Inequalities for Two Numbers ... - Claudi Alsina
CLAUDI ALSINA. Universitat Politècnica de Catalunya. ROGER B. NELSEN. Lewis & Clark College p,q ÷ 0, p ' q = 1 ÷. 1 p. ' 1 q. ' 4 and p +. 1 p. Ã. ·fl. ⹠÷'. 2.
Theorem ([1]). Every even perfect number Np = 2pâ1(2p â 1) for prime p = 3 is congruent to 1 or 6 modulo 7. In particular, p â¡ 1 mod 3 =â Np â¡ 1 mod 7 and p ...
Theorem. Every even perfect number Np = 2pâ1(2p â 1) with p ⥠3 prime is the sum of the first n odd cubes for n = 2(pâ1)/2, i.e., Np = 13 + 33 +···+ (2n â 1)3 [1].
Proof Without Words: Square Triangular Numbers and Almost. Isosceles Pythagorean Triples. Roger B. Nelsen ([email protected]), Lewis & Clark College, ...
Every even perfect number, Np = 2pâ1(2p â 1) with p ⥠3 prime, ... T. T p= = 2 â1. 3 +1 p n. T. T. 3 +1 n. = 1 + 9 n. Note that for p odd, 2p â 2 â¡ (â1)p + 1 â¡ 0 ...
a + bx, squaring to obtain x2 = a + bx, and solving for the positive root. An alternative method begins by dividing the quadratic by x to obtain x = b + a/x. Theorem.
Proof Without Words: The Golden Ratio. Roger B. Nelsen ([email protected]), Lewis & Clark College, Portland, OR. Theorem. If x > 0 and x = 1 + 1/x, then x = Ï ...
We prove wordlessly the arithmetic mean-geometric mean inequality for two positive numbers by an equivalent trigonometric inequality. Reference. 1. L. Tan, Proof without words: Eisenstein's duplication formula, Math. Mag. 71 (1998) 207, http:// · dx.
Proof Without Words: A Surprising Integer Result. Roger B. Nelsen ([email protected]), Lewis & Clark College, Portland, OR. Theorem. 48 = 47. Proof. Corollary ...
2. M. Benito and J. L. Varona, Advances in aliquot sequences, Math. Comp. 68 (1999), 389â393. ... (2002), Art. 52, 9pp. [http://jipam.vu.edu.au/v3n4/043 02.html].
Proof Without Words: A Sine Identity for Triangles. Roger B. Nelsen ([email protected]), Lewis & Clark College, Portland OR. If x + y + z = Ï, then 4(sin x)(sin ...
Let Tk denote the kth triangular number, Tk = 1 + 2 +···+ k. We show that. T3 + T6 +···+ T3n = 3(n + 1)Tn. Proof. I. T3k = 3(k2 + Tk). k. 2. T k. 3k k. 2 k. 2. T k. T k. II. n.
drowning on his left, even though saving the several would have been as easy as, and .... practical reason, but an account of a special category of reasons, ... rate not better. Moreover ..... to think that numbers count only where that brings a high
Available online xxxx. Keywords: ... massive parallel system which processes many elements ...... Analysis of the error rates on a subset of trials calling for.
The inequality is exact and the optimal values of c and k are given explicitly. It improves Kwapien's inequality in the case of the. Rademacher series. We also provide a new and very short proof of the Littlewood-Offord problem without using Sperner'
For each measure Z on Snâ1 the convex body Zâ â B may be defined as the convex body whose support function is given by. hZâ (u) = max{ u · v : v â supp ...
by Hansmann and Katriel [18] using the complex analytic approach developed in [1]. Their non-selfadjoint version of the LiebâThirring inequalities takes the.
Jul 23, 2011 - if Ïn is the volume of a unit ball in Rn, then. nnÏnVol(D)nâ1 ⤠Vol(âD)n and equality holds if and only if D is a ball. As an extension of the above classical isoperimetric inequality, it is conjectured that any n-dimensional c
Introduction. We study the following constrained infinite relaxation of a mixed-integer program: x =f + â ... pair (Ï ,Ï ) distinct from (Ï,Ï) such that Ï â¤ Ï and Ï â¤ Ï. ... function pair for Mf,S . (Ï,Ï) is minimal for Mf,S if and on
