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(/usr/share/texmf-tetex/tex/latex/base/omscmr.fd File: omscmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions ) LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <10> not available (Font) Font shape `OMS/cmsy/m/n' tried instead on input line 64. Overfull \hbox (2.86171pt too wide) in paragraph at lines 95--97 [][]\OT1/cmr/m/n/10 Write a program that enumerates rational right triangles (m ake table). [] [3] Overfull \hbox (0.50055pt too wide) in paragraph at lines 121--124 [][]\OT1/cmr/m/n/10 Use the internet to try to figure out what the ``local-to-g lobal principle [] Overfull \hbox (3.28244pt too wide) in paragraph at lines 124--131 [][]\OT1/cmr/bx/n/10 Open problem:[] \OT1/cmr/m/n/10 Consider an equation $\OML /cmm/m/it/10 y[] \OT1/cmr/m/n/10 = \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/ m/it/10 x\OT1/cmr/m/n/10 )$ with $\OML/cmm/m/it/10 f$ \OT1/cmr/m/n/10 a polynom ial [] Overfull \hbox (0.6302pt too wide) in paragraph at lines 143--145 [][]\OT1/cmr/m/n/10 Why are they called ``congruent numbers''? Connection betwe en congruent [] [4] [5] Overfull \hbox (7.69508pt too wide) in paragraph at lines 237--241 []\OT1/cmr/m/n/10 and read about another difficult generalization of the congru ent number [] File: graphics/elliptic_curves.png Graphic file (type eps) Overfull \hbox (71.87488pt too wide) in paragraph at lines 246--247 [][] [] [6] [7] Overfull \hbox (13.51134pt too wide) in paragraph at lines 259--264 [][]\OT1/cmr/m/n/10 (40 minutes) \OT1/cmr/m/it/10 Bijection \OT1/cmr/m/n/10 bet ween points and congruent triangles: revisited. [] Overfull \hbox (2.50053pt too wide) in paragraph at lines 281--286 \OT1/cmr/m/n/10 This is the intersection of a quadratic surface and a plane, he nce a quadratic [] Overfull \hbox (7.87099pt too wide) in paragraph at lines 291--295 \OT1/cmr/m/n/10 (Exercise -- 30 minutes doing this (and getting new triangles); presentations.) [] [8] Overfull \hbox (25.19734pt too wide) in paragraph at lines 373--373 []\OT1/cmtt/m/n/10 sage: print (4*n^2 + X^4).subst(Y,x_1+y_1^2).subst(X, y_1)[ ] [] [9] Overfull \hbox (9.64247pt too wide) in paragraph at lines 393--396 [][]\OT1/cmr/m/n/10 Show that the point on the cubic $\OML/cmm/m/it/10 y[] \OT1 /cmr/m/n/10 = \OML/cmm/m/it/10 x[] \OMS/cmsy/m/n/10 ^^@ \OT1/cmr/m/n/10 36\OML/ cmm/m/it/10 x$ \OT1/cmr/m/n/10 that corresponds [] [10] [11] Overfull \hbox (9.5601pt too wide) in paragraph at lines 494--497 [][]\OT1/cmr/m/n/10 (30 minutes) Participants: Graph elliptic curves. Then []\O T1/cmr/bx/n/10 derive an algebraic [] [12] Overfull \hbox (32.74715pt too wide) in paragraph at lines 553--553 []\OT1/cmtt/m/n/10 Elliptic Curve defined by y^2 = x^3 - 36*x over Rational F ield[] [] [13] Underfull \hbox (badness 10000) in paragraph at lines 589--594 [] Overfull \hbox (74.74678pt too wide) in paragraph at lines 629--629 []\OT1/cmtt/m/n/10 Elliptic Curve defined by y^2 = x^3 + 7*x over Finite Fiel d of size 43[] [] [14] File: graphics/swinnerton_dyer-count.png Graphic file (type eps) File: graphics/edsac.png Graphic file (type eps) [15] Underfull \hbox (badness 10000) in paragraph at lines 711--712 [] File: graphics/lser1.png Graphic file (type eps) File: graphics/lser6.png Graphic file (type eps) Overfull \hbox (32.2433pt too wide) in paragraph at lines 749--751 [][] $\OML/cmm/m/it/10 n \OT1/cmr/m/n/10 = 1$[] $[]\OML/cmm/m/it/10 n \OT1/cmr/ m/n/10 = 6$[] [] [16] File: graphics/lser34.png Graphic file (type eps) File: graphics/lser4199.png Graphic file (type eps) Overfull \hbox (52.24336pt too wide) in paragraph at lines 752--754 [][] $\OML/cmm/m/it/10 n \OT1/cmr/m/n/10 = 34$[] $[]\OML/cmm/m/it/10 n \OT1/cmr /m/n/10 = 4199$[] [] [17] Overfull \hbox (74.74678pt too wide) in paragraph at lines 786--786 []\OT1/cmtt/m/n/10 -0.0000000000000000... (the nonzeros at the end are just e rrors/noise)[] [] Overfull \hbox (43.24706pt too wide) in paragraph at lines 798--798 [] \OT1/cmtt/m/n/10 lval = 2*sum(v[n]/n*exp(-2*pi*n/sqrtN) for n in range(1, prec))[] [] Overfull \hbox (43.24706pt too wide) in paragraph at lines 834--834 [] \OT1/cmtt/m/n/10 return plot(EllipticCurve(GF(p),[-n^2,0]), rgbcolor=hue (0.7))[] [] Overfull \hbox (11.74733pt too wide) in paragraph at lines 834--834 []\OT1/cmtt/m/n/10 Q = [[P[4*i+j] for j in range(4)] for i in range(len(P)/4)] [] [] Overfull \hbox (37.9971pt too wide) in paragraph at lines 834--834 []\OT1/cmtt/m/n/10 # show(graphics_array(Q), axes=False) # if you don't want axes[] [] File: graphics/points_mod_p.png Graphic file (type eps) Overfull \hbox (48.68738pt too wide) in paragraph at lines 835--836 [] [] [] [18] Overfull \hbox (3.48799pt too wide) in paragraph at lines 855--857 [][]\OT1/cmr/m/n/10 Are there \OT1/cmr/m/it/10 any \OT1/cmr/m/n/10 integers $\O ML/cmm/m/it/10 n \OMS/cmsy/m/n/10 ^^Q \OT1/cmr/m/n/10 3([]8)$ with $\OML/cmm/m/ it/10 n$ \OT1/cmr/m/n/10 a congruent number? [] [19] Overfull \hbox (14.91722pt too wide) in paragraph at lines 881--883 [][]\OT1/cmr/m/n/10 Find Andrew Wiles's paper on the Birch and Swinnerton-Dyer conjecture [] Overfull \hbox (2.0949pt too wide) in paragraph at lines 886--887 [][]\OT1/cmr/m/n/10 ``Mathematicians have always been fascinated by the problem [] Overfull \hbox (3.55081pt too wide) in paragraph at lines 886--887 \OT1/cmr/m/n/10 of describing all solutions in whole numbers $\OML/cmm/m/it/10 x; y; z$ \OT1/cmr/m/n/10 to algebraic [] [20] File: graphics/pythag.png Graphic file (type eps) Overfull \hbox (23.49991pt too wide) in paragraph at lines 961--962 [][] [] [21] [22] Overfull \hbox (31.14445pt too wide) detected at line 1035 \OT1/cmr/m/n/10 # [] = []# [] \OML/cmm/m/it/10 : [] Overfull \hbox (35.2111pt too wide) detected at line 1041 \OT1/cmr/m/n/10 # [] = []# [] \OML/cmm/m/it/10 : [] [23] [24] [25] [26] Overfull \hbox (14.87273pt too wide) in paragraph at lines 1155--1155 [][]\OT1/cmr/bx/n/14.4 An Elliptic Curve Cryptography (ECC) Tutorial [] Overfull \hbox (2.25055pt too wide) in paragraph at lines 1172--1177 [][]\OT1/cmr/m/n/10 Elliptic curves are also used in several integer factorizat ion algorithms [] File: graphics/koblitz.png Graphic file (type eps) [27] Overfull \hbox (11.99689pt too wide) in paragraph at lines 1232--1232 []\OT1/cmtt/m/n/10 sage: E = EllipticCurve(F, [F.random_element(), F.random_ele ment()])[] [] Overfull \hbox (64.49643pt too wide) in paragraph at lines 1232--1232 []\OT1/cmtt/m/n/10 Elliptic Curve defined by y^2 = x^3 + 45*x + 43 over Finite Field of size 137[] [] [28] Overfull \hbox (11.99689pt too wide) in paragraph at lines 1276--1276 []\OT1/cmtt/m/n/10 sage: E = EllipticCurve(F, [F.random_element(), F.random_ele ment()])[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 1289--1289 []\OT1/cmtt/m/n/10 sage: E = EllipticCurve(F, [F.random_element(), F.random_ele ment()])[] [] [29] Overfull \hbox (94.74641pt too wide) in paragraph at lines 1336--1336 []\OT1/cmtt/m/n/10 Elliptic Curve defined by y^2 = x^3 + 508*x + 42 over Fini te Field of size 577[] [] [30] [31] (./notes.aux) ) Here is how much of TeX's memory you used: 3384 strings out of 95332 37124 string characters out of 1187736 102719 words of memory out of 1000000 6402 multiletter control sequences out of 10000+50000 15355 words of font info for 59 fonts, out of 500000 for 2000 0 hyphenation exceptions out of 8191 31i,12n,40p,239b,401s stack positions out of 1500i,500n,5000p,200000b,5000s Output written on notes.dvi (31 pages, 95804 bytes).