Content-Type: multipart/mixed; boundary="-------------0408020703757" This is a multi-part message in MIME format. ---------------0408020703757 Content-Type: text/plain; name="04-238.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-238.keywords" Nonlinear Schroedinger equation, Periodic solutions, Resonances, Lindstedt series, Renormalization Group, Counterterms ---------------0408020703757 Content-Type: application/postscript; name="nls.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="nls.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: nls.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX10 CMBX12 CMSY7 CMMI7 CMR10 CMCSC10 CMTI10 CMMI10 %%+ CMEX10 CMSY10 CMR7 CMR5 MSBM7 CMSY5 MSBM10 MSBM5 CMMI5 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o nls.ps nls %DVIPSParameters: dpi=600, compressed 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM5 %!PS-AdobeFont-1.1: MSBM5 2.1 %%CreationDate: 1992 Oct 17 08:36:07 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM5) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 86 /V put dup 87 /W put readonly def /FontBBox{0 -574 3027 1074}readonly def /UniqueID 5032013 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8B731A88D7740B0DDAED1B3F82 7DB9DFB4372D3935C286E39EE7AC9FB6A9B5CE4D2FAE1BC0E55AE02BFC464378 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%!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBAF552B11EFFB6A16C F03FB920C15AE724EFDF0CCBF00A838D34440FF9FED532F44036AD22561184C5 283722DDFA7285E62754372D716D704AC0E00B2F6AB67154241C7449AA047833 94CEDB08E8C92907FE72A0B05AE36A7B9226ACD6E7890A0B528FDDE84A950FC6 801DE75CF2E739E9121149CCB8B1C87A106822648D84A3D3FBF295EE6C4BF403 BBE9A1C1F6DAEDD1E642ACC486E609703D7612BFFD10C324F5DC710811F7F614 3691B400E3773987424C0D2B0D8A736873C6371DDB2442F05E018A2B5CA9A4AA 17AABB95D09E5890CFFFED5AC01495D89A53D3CF44556D089203F8FC8E7F9119 8496FAB6BEDB6FEE8621F38B59D1E356A63E388FF9CAF232FC7D21A03B2D5C44 EB73732C7283699CC4779260C1251E9D746F6769B64EA01EE3526E791803F0F2 75D012F19797FE2634E3CAAB160429E70359F425A17B6A5D07E1481AE000F5E0 680CCA88C092958B27086A3D4A585D57C557C81900FC6973046F8FA060D4E19A 7D2947019EC8497F582C319048C14802C58590DE66C3B6600247E316DC3CEDB7 61EAAFB0E8407EE735248D5FFE98EA18D62129CDC43BAE499C4D5A89029FD4A1 9A9FFEDA0CCBD329C32DDB310B8E9EB3D8A66B4F6664036F8AC05EE270A26092 8A8CCC25EC78D606DF23D2E8218831B466956A575913C1F5729AFC045609B145 CE7E6A710CE09481030187EDAC2B2581A2FDB317C64039A5E16F12481A205692 B8550F76A04723F40D4A472E5B973132ABDFFD159ED17B361FE3E93BF8336925 E9F902ED94D8F39AC9D7995E59DAF376299193F56E140C70064842227181EB90 54F8F40A00086E74A3DB50952B661BA32A0A95B44A239D66DF4E58B165674ADF 9A68885271091DB59B3971304DC4538FB16BF3B1C0452FFC752D49425B81ECC4 0DCAB13FF689C1CFE11347E9490BA9A0EEDA3D8BE0E2E5FE48CDB65473B2F83D 36CC0FD6A871BB6DC06AD79F8A97A5FE833D1F49532C9898864EF59F9F6D3322 E7035702F9127342536502B1500A0E62A8B9C6BFDED2815B8EEDCFFC1B4F41A9 9744D9E4FF7F3D8DC740A9A66D22FEAB0FF6C3CA47BB3B72A212A94BB9A80F53 702E62F7617F95C274EC82A4C9E4FB658D029DA392B987EE4340BA9B3969632B 563AB152F908DBD21C3B8CF42EC926755FCCA7D5F554E443E452F42FC1C63EF6 0488209510A4C97700A13B430EF35D018539F191817117FE9926E22DE04308D7 30F5A91D0B50405A754F5C9C9645971876D0E4081D43376FC1958F3D9DEA1095 1F816E92623ED3ECE261B49981DB09E8EE995A4FA3627B6CD56D48B8B53AFB9E 6575E40E03172E3BFC408B28939FF1F976DFA5492F3C76572DF5E1D48A557135 9ADF5143E089668F705EB9FD5E7626FED9039B41B1F6345FD02474C3D2FB67F8 BBD3B39175248920F53C6761C95F979189AA63AC110E49774FC13E 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec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D3196EBDFC659BF2993439A2318B5447AE5CE127A8546DBF0F18CDAF3C9E9831 84218980AE069738F894873E5AC00D438227D2C56E3E5EEE1E02788C997EC921 84F0B1E11D9B9058E49F296D0750279D24632F8F54CEEE832C8ADEAF52F853AD B9BECCB1F43376412F7FAE92AFD8CB60E431C5268BF12335DDD032295D91ECD0 EC6D391E8CCCDB37E00D4852950E943BE94CBB5B622FD8D78A77DC157BB5D1BF E2651D43FAEEB23C2AC22C39A5FA19ECB4D79D6324262453DEB7E9484751D43F A0B256AF0FF5AA14687CC3591C9E1446102C5949FF723167B2CB18F251DA892B B631FCCEAADE957757AFBA527146E95A9310405EF81A5304DB6A0C793BA43629 817DE642DDD2EF40BCA1EEF567AA9394C628BCD17A74AB699A59EC954D5D9DF5 B4BEF29C6C149BB6605D2E9BCB4FDC63E97859122DDB9E7A17DB6D252849ECE6 6FEF67B23D37CAC160B9A1A4D78198C2A109DC2D833D49273BF84EB274A989FA 6C7357AECB198C4AD697F7B767425C3183D87E8AE31D93F7C2BC1E590CABA638 52687E81A11937E973088AC02F87D97F7A8D944EB4D526E09E514C8CAB453392 1E0AF4ECF263BFDC329C12048FD655BF0D9CA9DB9C30 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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95060D5D07B4FB75692876F8BBD5BAA9D0E7C14249D9BB7C6FA13605E4BA1AF2 2632733C75C0946A1C66FD31C7DF84CE8E8FD5F6130704C2D4EC0C2AECFFC309 E4C59AF2A2AAA4826D1E7E9E467A35FD124C248164159750A00829A38E298393 39BEB961292FE2E0950B6AA984CCCFE06B5CBA618F14C3C8847AF45971534A44 D5454B932CB2AC58F51CF78267390B174FC49C5FB383D41F2FEB849A31B03400 78C0F1D483AE3C494FA96BD50E638AC6633222B75FE7325E46A833B1F09A5955 171950FB5BB4AA8F82792A5234E5863000213FADEA23FA348391470B1B043A29 BBF90BAC2136807153BA2A6C219A56EB3EC096599AD8C35F7BFEB77F4379917E 5387F5699E8901428D62C500FA2F3E9C835236007DC2E3CAE8FFF2CC08968C4F F2359E0BD684F353F06A99A80D0BF9D1AB7BDE8F204E0EA867A8B9105F83201D 6B3B128481FFBD1BB7CF9100FE7D35DF4B6D51816BC709CA92145094EB32340D B6C97F2AE9A9B0E4096CB6F5244577DBBB6FCA7E87A54F2D2C7C60F209D47657 4C3762E8C3D21B883B3D245DDD322FD824A1BDD085BE449F29C6C96F9F9FF9D1 DC120D795F71EF6467AA63A309B30336A46AED0EFCD3D55BBDB0A9A426528504 1488AF4F63E3D616FD47AC874114BE044551B5D06F4526A58685877D4EFE2782 77D47AC9EED2A3052209C7770FA47CBB7BF9E7B3D993C7324D89856E57588430 C3679DE014A1701B89A0F980BF69A04E17A08C41305047246AF48637F0B65671 FB17F3C00B3CF404AC1BEF1373923EFCBD4524D82E4ABFBCB89F9853BC55BBA5 22B1057A3EA689A2A630913FFD91FD7E2CCD64F048B3E3AA3E9EFFFC4A7BD97B 25B2B3C5104CBE84EA3C4EB1B4F34A2B30F1BFCEA5CDA57D7E4F64DD7292E059 BD8F9137455073ECCDAC455E07B38FCC78180D0E35994F 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 26 /braceleftbigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 80 /summationtext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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5167 y Fj(\()p Fq(b)p Fj(\))1793 5222 y Fq(m)1884 5201 y Fm(w)1943 5213 y Fq(n;)p Fr(\000)p Fq(m)2137 5201 y Fp(+)g Fm(")p Fp([)p Fm(')p Fp(\()p Fm(v)k Fp(+)c Fm(w)r Fp(\)])2629 5213 y Fq(n;m)2754 5201 y Fm(;)3184 5130 y Fp(\(2)p Fm(:)p Fp(2\))1680 5396 y(3)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 47 83 a Fp(where)1396 185 y Fm(\027)1442 150 y Fj(\()p Fq(a)p Fj(\))1437 205 y Fq(m)1553 185 y Fk(\000)18 b Fm(\027)1682 150 y Fj(\()p Fq(b)p Fj(\))1677 205 y Fq(m)1791 185 y Fp(=)k Fm(\027)1919 197 y Fq(m)1982 185 y Fm(:)1179 b Fp(\(2)p Fm(:)p Fp(3\))118 337 y(With)29 b(the)f(notations)e(of)i(\(1.12\),)f(and)g(recalling)g(that)g(w)n(e)h (are)e(considering)h Fm(')p Fp(\()p Fm(x)p Fp(\))d(=)f Fm(x)p Fp(,)28 b(w)n(e)f(ha)n(v)n(e)181 452 y Fl(\002)216 520 y Fk(j)p Fm(v)22 b Fp(+)c Fm(w)r Fk(j)468 485 y Fj(2)506 520 y Fp(\()p Fm(v)k Fp(+)c Fm(w)r Fp(\))776 452 y Fl(\003)811 553 y Fq(m)897 520 y Fp(=)23 b([)p Fk(j)p Fm(v)s Fk(j)1097 485 y Fj(2)1135 520 y Fm(v)s Fp(])1201 532 y Fq(m)1282 520 y 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Fm(x)p Fp(\),)27 b(so)g(that)1046 1139 y Fm(b)p Fp(\()p Fm(t;)14 b(x)p Fp(\))23 b(=)g Fm(B)t(e)1477 1105 y Fq(im)1559 1080 y Fi(2)1559 1121 y(0)1592 1105 y Fq(t)p Fr(\000)p Fq(im)1751 1113 y Fi(0)1783 1105 y Fq(x)1825 1139 y Fm(;)180 b(B)27 b Fp(=)c Fk(\000)p Fm(A;)828 b Fp(\(2)p Fm(:)p Fp(5\))47 1323 y(so)27 b(that)h(w)n(e)f(obtain)410 1485 y([)p Fk(j)p Fm(v)s Fk(j)522 1451 y Fj(2)560 1485 y Fm(v)s Fp(])626 1497 y Fq(m)712 1485 y Fp(=)c Fk(j)p Fm(A)p Fk(j)908 1451 y Fj(2)946 1485 y Fm(A\016)1045 1497 y Fq(m;m)1183 1505 y Fi(0)1237 1485 y Fp(+)18 b Fk(j)p Fm(B)t Fk(j)1433 1451 y Fj(2)1471 1485 y Fm(B)t(\016)1575 1497 y Fq(m;)p Fr(\000)p Fq(m)1765 1505 y Fi(0)1819 1485 y Fp(+)g(2)p Fk(j)p Fm(A)p Fk(j)2052 1451 y Fj(2)2090 1485 y Fm(B)t(\016)2194 1497 y Fq(m;)p Fr(\000)p Fq(m)2384 1505 y Fi(0)2438 1485 y Fp(+)g(2)p Fk(j)p Fm(B)t Fk(j)2676 1451 y Fj(2)2713 1485 y Fm(A\016)2812 1497 y Fq(m;m)2950 1505 y Fi(0)708 1621 y Fp(+)g(2)p Fk(j)p Fm(A)p Fk(j)941 1586 y Fj(2)978 1621 y Fm(V)1026 1633 y Fq(m)1108 1621 y Fp(+)g(2)p Fk(j)p Fm(B)t Fk(j)1346 1586 y Fj(2)1383 1621 y Fm(V)1431 1633 y Fq(m)1513 1621 y Fp(+)g(2)p 1638 1554 63 4 v Fm(AB)t(V)1815 1633 y Fr(\000)p Fq(m)1930 1621 y Fm(\016)1967 1633 y Fq(m;)p Fr(\000)p Fq(m)2157 1641 y Fi(0)2212 1621 y Fp(+)g(2)p 2337 1554 68 4 v Fm(B)s(AV)2513 1633 y Fr(\000)p Fq(m)2629 1621 y Fm(\016)2666 1633 y Fq(m;m)2804 1641 y Fi(0)708 1756 y Fp(+)p 791 1690 67 4 v 18 w Fm(V)858 1768 y Fq(m)921 1756 y Fm(A)983 1722 y Fj(2)1020 1756 y Fm(\016)1057 1768 y Fq(m;m)1195 1776 y Fi(0)1250 1756 y Fp(+)g(2)p 1375 1690 V Fm(V)1441 1768 y Fr(\000)p Fq(m)1556 1756 y Fm(AB)t(\016)1722 1768 y Fq(m;)p Fr(\006)p Fq(m)1912 1776 y Fi(0)1967 1756 y Fp(+)p 2050 1690 V 18 w Fm(V)2117 1768 y Fq(m)2180 1756 y Fm(B)2247 1722 y Fj(2)2284 1756 y Fm(\016)2321 1768 y Fq(m;)p Fr(\000)p Fq(m)2511 1776 y Fi(0)2566 1756 y Fp(+)g([)p Fm(G)2737 1768 y Fj(1)2774 1756 y Fp(\()p Fm(v)s Fp(\)])2904 1768 y Fq(m)2968 1756 y Fm(;)3184 1620 y Fp(\(2)p Fm(:)p Fp(6\))47 1938 y(where)27 b Fm(G)352 1950 y Fj(1)390 1938 y Fp(\()p Fm(v)s Fp(\))h(is)g(at)f(least)g(quadratic)g(in)h Fm(V)19 b Fp(.)118 2040 y(Then,)32 b(b)n(y)e(setting)g Fm(G)p Fp(\()p Fm(v)s(;)14 b(w)r Fp(\))30 b(=)d Fm(G)1212 2052 y Fj(1)1249 2040 y Fp(\()p Fm(v)s Fp(\))22 b(+)e Fm(G)1528 2052 y Fj(2)1565 2040 y Fp(\()p Fm(v)s(;)14 b(w)r Fp(\),)33 b(the)e(Q)f(equation)f(in)i(\(1.11\))f(can)g(b)r(e)h(rewritten)47 2141 y(for)c Fm(m)c Fp(=)g Fm(m)431 2153 y Fj(0)496 2141 y Fp(as)203 2316 y Fm(m)276 2282 y Fj(2)276 2337 y(0)313 2316 y Fm(A)g Fp(=)g Fk(j)p Fm(A)p Fk(j)594 2282 y Fj(2)631 2316 y Fm(A)c Fp(+)f(2)p Fk(j)p Fm(B)t Fk(j)950 2282 y Fj(2)987 2316 y Fm(A;)121 2452 y(m)194 2418 y Fj(2)194 2472 y(0)231 2452 y Fm(V)279 2464 y Fq(m)338 2472 y Fi(0)398 2452 y Fp(=)23 b(2)p Fk(j)p Fm(A)p Fk(j)636 2418 y Fj(2)673 2452 y Fm(V)721 2464 y Fq(m)780 2472 y Fi(0)835 2452 y Fp(+)18 b(2)p Fk(j)p Fm(B)t Fk(j)1073 2418 y Fj(2)1110 2452 y Fm(V)1158 2464 y Fq(m)1217 2472 y Fi(0)1273 2452 y Fp(+)p 1356 2385 V 18 w Fm(V)1422 2464 y Fq(m)1481 2472 y Fi(0)1518 2452 y Fm(A)1580 2418 y Fj(2)1636 2452 y Fp(+)g(2)p Fm(AB)p 1890 2385 V 4 w(V)1957 2464 y Fr(\000)p Fq(m)2068 2472 y Fi(0)2122 2452 y Fp(+)h(2)p 2248 2385 68 4 v Fm(B)s(AV)2424 2464 y Fr(\000)p Fq(m)2535 2472 y Fi(0)2591 2452 y Fp(+)f([)p Fm(G)p Fp(\()p Fm(v)s(;)c(w)r Fp(\)])2990 2464 y Fq(m)3049 2472 y Fi(0)3087 2452 y Fm(;)3184 2383 y Fp(\(2)p Fm(:)p Fp(7\))47 2629 y(and)28 b(for)f(p)r(ositiv)n(e)g Fm(m)c Fk(6)p Fp(=)g Fm(m)900 2641 y Fj(0)964 2629 y Fp(as)930 2813 y Fm(m)1003 2779 y Fj(2)1040 2813 y Fm(V)1088 2825 y Fq(m)1175 2813 y Fp(=)f(2)p Fk(j)p Fm(A)p Fk(j)1412 2779 y Fj(2)1450 2813 y Fm(V)1498 2825 y Fq(m)1580 2813 y Fp(+)c(2)p Fk(j)p Fm(B)t Fk(j)1818 2779 y Fj(2)1855 2813 y Fm(V)1903 2825 y Fq(m)1985 2813 y Fp(+)g([)p Fm(G)p Fp(\()p Fm(v)s(;)c(w)r Fp(\)])2384 2825 y Fq(m)2448 2813 y Fm(;)713 b Fp(\(2)p Fm(:)p Fp(8\))47 2997 y(while)20 b(the)g(equation)f(for)g(negativ)n(e)f (v)-5 b(alues)19 b(of)h Fm(m)f Fp(can)g(b)r(e)h(obtained)f(b)n(y)g (using)g(the)h(symmetry)f(prop)r(erties)47 3099 y(\(1.10\),)27 b(whic)n(h)h(imply)g Fm(V)829 3111 y Fr(\000)p Fq(m)967 3099 y Fp(=)23 b Fk(\000)p Fm(V)1168 3111 y Fq(m)1231 3099 y Fp(.)118 3200 y(By)k(de\014ning)h Fm(\013)c Fp(=)e Fk(j)p Fm(A)p Fk(j)834 3170 y Fj(2)895 3200 y Fp(=)p 982 3133 63 4 v 22 w Fm(A)q(A)28 b Fp(and)f(using)g(the)h(iden)n (tities)446 3384 y Fm(\013)23 b Fp(=)p 610 3317 V 23 w Fm(AA)h Fp(=)p 845 3317 68 4 v 22 w Fm(B)t(B)k Fp(=)22 b Fk(\000)p 1155 3317 63 4 v Fm(AB)27 b Fp(=)c Fk(\000)p 1460 3317 68 4 v Fm(B)t(A;)180 b(\014)27 b Fp(=)c Fm(AA)g Fp(=)g Fm(B)t(B)k Fp(=)c Fk(\000)p Fm(AB)k Fp(=)c Fk(\000)p Fm(B)t(A;)228 b Fp(\(2)p Fm(:)p Fp(9\))47 3568 y(whic)n(h)25 b(follo)n(w)g(trivially)g(from)g(the)g(de\014nitions)h(of)f Fm(A)h Fp(and)f Fm(B)k Fp(in)d(\(1.5\))f(and)g(\(2.5\))g(resp)r(ectiv)n (ely)-7 b(,)25 b(w)n(e)g(can)47 3669 y(rewrite)i(\(2.7\))g(as)654 3853 y Fm(m)727 3818 y Fj(2)727 3873 y(0)764 3853 y Fm(A)c Fp(=)g(3)p Fm(\013A;)572 3988 y(m)645 3954 y Fj(2)645 4009 y(0)682 3988 y Fm(V)730 4000 y Fq(m)789 4008 y Fi(0)849 3988 y Fp(=)g(4)p Fm(\013V)1080 4000 y Fq(m)1139 4008 y Fi(0)1194 3988 y Fp(+)18 b Fm(\014)p 1328 3922 67 4 v 4 w(V)1395 4000 y Fq(m)1454 4008 y Fi(0)1509 3988 y Fk(\000)g Fp(2)p Fm(\014)p 1685 3922 V 4 w(V)1752 4000 y Fr(\000)p Fq(m)1863 4008 y Fi(0)1918 3988 y Fk(\000)g Fp(2)p Fm(\013V)2144 4000 y Fr(\000)p Fq(m)2255 4008 y Fi(0)2310 3988 y Fp(+)g([)p Fm(G)p Fp(\()p Fm(v)s(;)c(w)r Fp(\)])2709 4000 y Fq(m)2768 4008 y Fi(0)2806 3988 y Fm(;)3142 3920 y Fp(\(2)p Fm(:)p Fp(10\))47 4175 y(where)28 b(the)g(\014rst)g(equation)g(de\014nes)g Fm(\013)c Fp(=)f Fk(j)p Fm(A)p Fk(j)1489 4145 y Fj(2)1551 4175 y Fp(=)g Fm(m)1712 4145 y Fj(2)1712 4196 y(0)1749 4175 y Fm(=)p Fp(3.)38 b(By)28 b(using)f(once)h(more)f(the)i(iden)n(tities)f (\(1.10\))47 4277 y(and)36 b(imp)r(osing)g(that)g(the)h(co)r(e\016cien) n(ts)e Fm(V)1395 4289 y Fq(m)1495 4277 y Fp(b)r(e)h(real,)h(so)f(that)g Fm(\013)i Fp(=)e Fm(\014)41 b Fp(in)36 b(\(2.9\),)i(w)n(e)e(can)f (write)h(the)47 4379 y(second)27 b(equation)g(in)h(\(2.10\))f(and)g (the)h(equation)f(\(2.8\),)h(resp)r(ectiv)n(ely)-7 b(,)27 b(as)1046 4493 y Fl(\032)1122 4563 y Fm(m)1195 4533 y Fj(2)1195 4584 y(0)1232 4563 y Fm(V)1280 4575 y Fq(m)1339 4583 y Fi(0)1399 4563 y Fp(=)22 b(9)p Fm(\013V)1629 4575 y Fq(m)1688 4583 y Fi(0)1744 4563 y Fp(+)c([)p Fm(G)p Fp(\()p Fm(v)s(;)c(w)r Fp(\)])2143 4575 y Fq(m)2202 4583 y Fi(0)2240 4563 y Fm(;)1122 4663 y(m)1195 4633 y Fj(2)1232 4663 y Fm(V)1280 4675 y Fq(m)1366 4663 y Fp(=)23 b(4)p Fm(\013V)1597 4675 y Fq(m)1679 4663 y Fp(+)18 b([)p Fm(G)p Fp(\()p Fm(v)s(;)c(w)r Fp(\)])2078 4675 y Fq(m)2143 4663 y Fm(;)3142 4610 y Fp(\(2)p Fm(:)p Fp(11\))47 4838 y(so)27 b(that)h(w)n(e)f(\014nd)1175 4864 y Fl(8)1175 4938 y(>)1175 4963 y(<)1175 5113 y(>)1175 5138 y(:)1263 4956 y Fm(V)1311 4968 y Fq(m)1370 4976 y Fi(0)1430 4956 y Fp(=)c Fk(\000)1619 4899 y Fp(1)p 1593 4936 95 4 v 1593 5012 a(6)p Fm(\013)1697 4956 y Fp([)p Fm(G)p Fp(\()p Fm(v)s(;)14 b(w)r Fp(\)])2013 4968 y Fq(m)2072 4976 y Fi(0)2110 4956 y Fm(;)1263 5173 y(V)1311 5185 y Fq(m)1398 5173 y Fp(=)22 b Fk(\000)1566 5117 y Fp(1)p 1560 5154 54 4 v 1560 5230 a Fm(\013)1623 5173 y Fp([)p Fm(G)p Fp(\()p Fm(v)s(;)14 b(w)r Fp(\)])1939 5185 y Fq(m)2004 5173 y Fm(;)3142 5059 y Fp(\(2)p Fm(:)p Fp(12\))1680 5396 y(4)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 47 83 a Fp(resp)r(ectiv)n(ely)27 b(for)g(p)r(ositiv)n(e)g Fm(m)1007 95 y Fj(0)1072 83 y Fp(and)g Fm(m)c Fk(6)p Fp(=)g Fm(m)1490 95 y Fj(0)1527 83 y Fp(.)118 185 y(Finally)28 b(w)n(e)f(write)334 385 y Fm(w)393 397 y Fq(n;m)540 385 y Fp(=)c Fm(g)s Fp(\()p Fm(n;)14 b(m)p Fp(\))909 293 y Fl(\020)958 385 y Fm(\026\027)1054 351 y Fj(\()p Fq(a)p Fj(\))1049 406 y Fq(m)1147 385 y Fm(w)1206 397 y Fq(n;m)1348 385 y Fp(+)k Fm(\026\027)1527 351 y Fj(\()p Fq(b)p Fj(\))1522 406 y Fq(m)1613 385 y Fm(w)1672 397 y Fq(n;)p Fr(\000)p Fq(m)1866 385 y Fp(+)h Fm(\026")p Fp([)p Fk(j)p Fm(v)i Fp(+)d Fm(w)r Fk(j)2313 351 y Fj(2)2351 385 y Fp(\()p Fm(v)k Fp(+)c Fm(w)r Fp(\)])2644 397 y Fq(n;m)2769 293 y Fl(\021)2832 385 y Fm(;)287 b Fp(\(2)p Fm(:)p Fp(13\))47 586 y(where)1093 704 y Fm(g)s Fp(\()p Fm(n;)14 b(m)p Fp(\))23 b(=)1620 648 y(1)p 1481 685 321 4 v 1481 761 a Fm(!)s(n)18 b Fk(\000)24 b Fp(~)-48 b Fm(!)1742 737 y Fj(2)1739 782 y Fq(m)1811 704 y Fm(;)180 b(n)23 b Fk(6)p Fp(=)g Fm(m)2248 670 y Fj(2)2285 704 y Fm(;)834 b Fp(\(2)p Fm(:)p Fp(14\))47 898 y(and)28 b(w)n(e)f(lo)r(ok)g(for)g(a)g(solution)g Fm(u)1069 910 y Fq(n;m)1220 898 y Fp(in)h(the)g(form)f(of)h(a)f(p)r(o)n(w)n(er)f (series)h(expansion)g(in)g Fm(\026)p Fp(,)1350 1148 y Fm(u)1398 1160 y Fq(n;m)1544 1148 y Fp(=)1659 1044 y Fr(1)1632 1069 y Fl(X)1632 1247 y Fq(k)q Fj(=0)1766 1148 y Fm(\026)1816 1113 y Fq(k)1857 1148 y Fm(u)1905 1113 y Fj(\()p Fq(k)q Fj(\))1905 1168 y Fq(n;m)2029 1148 y Fm(;)1090 b Fp(\(2)p Fm(:)p Fp(15\))47 1432 y(with)28 b Fm(u)284 1389 y Fj(\()p Fq(k)q Fj(\))284 1441 y Fq(n;m)435 1432 y Fp(dep)r(ending)g(on)g Fm(")f 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5123 y Fj(0)p Fq(;m)2240 5111 y Fm(v)2280 5123 y Fj(0)p Fq(;m)2396 5111 y Fm(v)2436 5123 y Fj(0)p Fq(;m)2552 5111 y Fm(;)3184 4992 y Fp(\(4)p Fm(:)p Fp(2\))1680 5396 y(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 47 83 a Fp(so)27 b(that)h(w)n(e)f(obtain)1094 185 y Fm(v)1134 197 y Fj(0)p Fq(;m)1264 117 y Fl(\000)1302 185 y Fm(m)1375 150 y Fj(2)1431 185 y Fk(\000)18 b Fp(2)p Fk(k)p Fm(v)1638 197 y Fj(0)1674 185 y Fk(k)1716 150 y Fj(2)1772 185 y Fp(+)g Fk(j)p Fm(v)1918 197 y Fj(0)p Fq(;m)2034 185 y Fk(j)2057 150 y Fj(2)2094 117 y Fl(\001)2155 185 y Fp(=)23 b(0)p Fm(;)876 b Fp(\(4)p Fm(:)p Fp(3\))47 339 y(where)27 b(w)n(e)h(ha)n(v)n(e)e(de\014ned)670 528 y Fk(M)c Fp(=)h Fk(f)p Fm(m)f Fk(2)i Fh(Z)f Fp(:)h Fm(v)1252 540 y Fj(0)p Fq(;m)1391 528 y Fk(6)p Fp(=)e(0)p Fk(g)13 b Fm(;)180 b Fk(M)1878 540 y Fj(+)1956 528 y Fp(=)23 b Fk(f)o Fm(m)g Fk(2)h(M)e Fp(:)h Fm(m)g(>)g Fp(0)p Fk(g)13 b Fm(;)452 b Fp(\(4)p Fm(:)p Fp(4\))47 717 y(and)28 b(set)1144 819 y Fk(k)p Fm(v)1226 831 y Fj(0)1263 819 y 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Fp(\(4)p Fm(:)p Fp(7\))47 1972 y(By)28 b(inserting)f(\(4.6\))g(in)n(to)g(\(4.3\),)h(setting)731 2198 y Fm(a)775 2210 y Fq(m)861 2198 y Fp(=)23 b Fm(v)989 2210 y Fj(0)p Fq(;m)1105 2198 y Fm(;)97 b(m)23 b(>)g Fp(0)p Fm(;)179 b Fk(k)p Fm(a)p Fk(k)1781 2164 y Fj(2)1840 2198 y Fp(=)1982 2119 y Fl(X)1928 2297 y Fq(m)p Fr(2M)2110 2305 y Fi(+)2170 2198 y Fm(a)2214 2164 y Fj(2)2214 2218 y Fq(m)2300 2198 y Fp(=)2398 2142 y(1)p 2398 2179 42 4 v 2398 2255 a(2)2449 2198 y Fk(k)p Fm(v)2531 2210 y Fj(0)2568 2198 y Fk(k)2610 2164 y Fj(2)2647 2198 y Fm(;)514 b Fp(\(4)p Fm(:)p Fp(8\))47 2479 y(and)28 b(writing)f Fk(M)595 2491 y Fj(+)673 2479 y Fp(=)22 b Fk(f)p Fm(m)875 2491 y Fj(1)912 2479 y Fm(;)14 b(m)1022 2491 y Fj(2)1059 2479 y Fm(;)g(:)g(:)g(:)g(;)g(m)1317 2491 y Fq(N)1379 2479 y Fk(g)p Fp(,)28 b(with)g Fm(m)1734 2491 y Fq(k)1798 2479 y Fm(<)22 b(m)1958 2491 y Fq(k)q Fj(+1)2083 2479 y Fp(,)28 b Fm(k)e Fp(=)c(1)p Fm(;)14 b(:)g(:)g(:)f(;)h(N)27 b Fk(\000)19 b Fp(1,)27 b(w)n(e)g(obtain)211 2705 y Fm(a)255 2671 y Fj(2)255 2726 y Fq(m)314 2735 y Fb(k)378 2705 y Fp(=)22 b(4)p Fk(k)p Fm(a)p Fk(k)635 2671 y Fj(2)690 2705 y Fk(\000)c Fm(m)846 2671 y Fj(2)846 2726 y Fq(k)909 2705 y Fp(=)1116 2649 y(4)p 1007 2686 261 4 v 1007 2762 a(4)p Fm(N)27 b Fk(\000)18 b Fp(1)1291 2638 y Fl(\000)1329 2705 y Fm(m)1402 2671 y Fj(2)1402 2726 y(1)1458 2705 y Fp(+)g Fm(m)1614 2671 y Fj(2)1614 2726 y(2)1669 2705 y Fp(+)g Fm(:)c(:)g(:)19 b Fp(+)f Fm(m)2024 2671 y Fj(2)2024 2726 y Fq(N)2086 2638 y Fl(\001)2143 2705 y Fk(\000)g Fm(m)2299 2671 y Fj(2)2299 2726 y Fq(k)2340 2705 y Fm(;)180 b(k)26 b Fp(=)c(1)p Fm(;)14 b(:)g(:)g(:)f(;)h(N)t(;)165 b Fp(\(4)p Fm(:)p Fp(9\))47 2923 y(whic)n(h)28 b(mak)n(es)e(sense)h(as) g(long)g(as)1171 3144 y(max)1134 3198 y Fq(m)p Fr(2M)1316 3206 y Fi(+)1377 3144 y Fm(m)1450 3110 y Fj(2)1510 3144 y Fk(\024)1717 3088 y Fp(4)p 1608 3125 V 1608 3201 a(4)p Fm(N)f Fk(\000)18 b Fp(1)1946 3065 y Fl(X)1892 3243 y Fq(m)p Fr(2M)2074 3251 y Fi(+)2134 3144 y Fm(m)2207 3110 y Fj(2)2244 3144 y Fm(:)875 b Fp(\(4)p Fm(:)p Fp(10\))47 3420 y(The)28 b(follo)n(wing)e(result)i(is)f(easily)g(pro)n(v)n(ed.)47 3593 y Fe(Lemma)32 b(5.)k Fn(F)-6 b(or)30 b(al)t(l)h Fm(N)h Fk(\025)22 b Fp(2)29 b Fn(ther)l(e)h(ar)l(e)g(solutions)g(of)h (\(4.9\))g(such)f(that)f Fp(4)p Fk(k)p Fm(a)p Fk(k)2558 3563 y Fj(2)2624 3593 y Fn(is)h(not)f(an)h(inte)l(ger.)47 3765 y(Pr)l(o)l(of.)39 b Fp(T)-7 b(o)27 b(obtain)f(a)h(solution)g(one)g (can)g(tak)n(e)f Fm(m)1621 3777 y Fq(k)1685 3765 y Fp(=)d Fm(m)1846 3777 y Fq(N)1926 3765 y Fk(\000)18 b Fp(\()p Fm(N)26 b Fk(\000)18 b Fm(k)s Fp(\))27 b(for)g Fm(k)f Fp(=)c(1)p Fm(;)14 b(:)g(:)g(:)f(;)h(N)9 b Fp(,)28 b(and)f(c)n(ho)r (ose)47 3867 y Fm(m)120 3879 y Fq(N)206 3867 y Fk(\025)c Fp(4)p Fm(N)9 b Fp(\()p Fm(N)25 b Fk(\000)16 b Fp(1\).)36 b(Cho)r(ose)26 b Fm(m)1111 3879 y Fq(N)1197 3867 y Fp(=)c(\(4)p Fm(N)j Fp(+)16 b Fm(j)5 b Fp(\)\()p Fm(N)26 b Fk(\000)16 b Fp(1\),)27 b(with)g Fm(j)h Fk(2)23 b(f)p Fp(0)p Fm(;)14 b Fp(1)p Fk(g)p Fp(:)34 b(then)28 b(4\()p Fm(m)2858 3837 y Fj(2)2858 3888 y(1)2911 3867 y Fp(+)16 b Fm(:)e(:)g(:)i Fp(+)g Fm(m)3259 3837 y Fj(2)3259 3890 y Fq(N)3322 3867 y Fp(\))47 3969 y(can)27 b(not)h(b)r(e)g(a)f(m)n(ultiple)h(of)g(4)p Fm(N)f Fk(\000)18 b Fp(1)27 b(for)g(b)r(oth)h Fm(j)g Fp(=)23 b(0)k(and)g Fm(j)h Fp(=)23 b(1.)p 3313 3961 42 42 v 118 4141 a(Here)d(w)n(e)f(are)g(con\014ned)h(ourselv)n(es)e(only)i (to)g(an)g(existence)f(result.)35 b(Of)20 b(course)e(more)i(general)e (solutions)47 4243 y(can)33 b(b)r(e)g(en)n(visaged,)f(with)h(more)f (spacing)g(b)r(et)n(w)n(een)h(the)g(in)n(v)n(olv)n(ed)e(w)n(a)n(v)n(e)g (n)n(um)n(b)r(ers)h Fm(m)2859 4255 y Fq(k)2900 4243 y Fp(.)52 b(The)33 b(result)47 4344 y(ab)r(o)n(v)n(e)26 b(can)i(indeed)g(b)r(e)g(strengh)n(tened)e(as)h(follo)n(ws.)47 4517 y Fe(Lemma)h(6.)35 b Fn(F)-6 b(or)26 b(al)t(l)i Fm(N)j Fk(\025)23 b Fp(2)j Fn(and)h(for)g(al)t(l)h(incr)l(e)l(asing)f (lists)f(of)h(p)l(ositive)h(inte)l(gers)f Fm(I)j Fp(:=)22 b Fk(f)p Fm(i)2915 4529 y Fj(1)2952 4517 y Fm(;)14 b(:)g(:)g(:)f(;)h(i) 3165 4529 y Fq(N)6 b Fr(\000)p Fj(1)3313 4517 y Fk(g)47 4618 y Fn(ther)l(e)29 b(exists)g Fm(m)553 4630 y Fq(N)615 4618 y Fp(\()p Fm(I)7 b Fp(\))30 b Fn(\()p Fm(m)859 4630 y Fq(N)950 4618 y Fn(for)g(short\))f(such)g(that)g(\(4.10\))h(has)g(a)f (solution)g(in)g(the)g(set)f Fk(M)2930 4630 y Fj(+)3008 4618 y Fp(=)23 b Fk(f)p Fm(m)3211 4630 y Fq(N)3290 4618 y Fk(\000)47 4720 y Fm(i)76 4732 y Fq(N)6 b Fr(\000)p Fj(1)224 4720 y Fm(;)14 b(:)g(:)g(:)f(;)h(m)481 4732 y Fq(N)563 4720 y Fk(\000)k Fm(i)675 4732 y Fj(1)711 4720 y Fm(;)c(m)821 4732 y Fq(N)884 4720 y Fk(g)29 b Fn(with)i Fp(4)p Fk(k)p Fm(a)p Fk(k)1306 4690 y Fj(2)1365 4720 y Fk(6)p Fp(=)22 b Fm(m)1525 4690 y Fj(2)1592 4720 y Fn(for)31 b(al)t(l)f Fm(m)j(=)-52 b Fk(2)24 b(M)p Fn(.)47 4892 y(Pr)l(o)l(of.)36 b Fp(Fix)20 b(the)f(set)h(of)f(in)n(tegers)g Fm(I)30 b Fp(=)22 b Fk(f)p Fm(i)1307 4904 y Fj(1)1344 4892 y Fm(;)14 b(:)g(:)g(:)f(;)h(i)1557 4904 y Fq(N)6 b Fr(\000)p Fj(1)1705 4892 y Fk(g)p Fp(,)20 b(and)g(consider)e(the)i (expression)e Fm(M)11 b Fk(\000)r Fp(\()p Fm(N)g Fk(\000)r Fp(1)p Fm(=)p Fp(4\))p Fm(j)3319 4862 y Fj(2)47 4994 y Fp(for)27 b Fm(j)h Fk(2)23 b Ff(N)p Fp(.)37 b(F)-7 b(or)26 b Fm(j)i Fp(=)23 b Fm(m)810 5006 y Fq(N)900 4994 y Fp(it)28 b(b)r(ecomes)e(a)h(p)r(olynomial)g(of)g(degree)f(t)n(w)n(o)g (in)h Fm(m)2487 5006 y Fq(N)2550 4994 y Fp(,)h(with)f(p)r(ositiv)n(e)g (leading)47 5095 y(co)r(e\016cien)n(t)36 b(1)p Fm(=)p Fp(4)f(and)h(p)r(ositiv)n(e)g(discriminan)n(t.)62 b(Hence)37 b(there)f(is)g(an)g(in)n(teger)f Fm(K)2673 5107 y Fj(1)2746 5095 y Fp(suc)n(h)h(that)h(for)f(all)47 5197 y Fm(m)120 5209 y Fq(N)206 5197 y Fm(>)23 b(K)365 5209 y Fj(1)429 5197 y Fp(one)k(has)g Fm(f)770 5209 y Fj(1)807 5197 y Fp(\()p Fm(m)912 5209 y Fq(N)976 5197 y Fp(\))c Fk(\021)g Fm(M)k Fk(\000)18 b Fp(\()p Fm(N)27 b Fk(\000)18 b Fp(1)p Fm(=)p Fp(4\))p Fm(m)1750 5167 y Fj(2)1750 5220 y Fq(N)1835 5197 y Fm(>)k Fp(0,)28 b(hence)f(\(4.10\))g(is)h(satis\014ed.)1659 5396 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 118 83 a Fp(The)27 b(inequalit)n(y)f Fm(M)f Fk(\000)16 b Fp(\()p Fm(N)26 b Fk(\000)16 b Fp(1)p Fm(=)p Fp(4\))p Fm(j)1262 53 y Fj(2)1321 83 y Fm(>)22 b Fp(0)27 b(is)f(trivially)g(satis\014ed)g(for)g Fm(j)i Fk(\024)23 b Fm(m)2539 95 y Fq(N)2602 83 y Fp(,)k(so)f(that)h(it)g(is)f (enough)47 185 y(to)31 b(lo)r(ok)g(an)g(in)n(teger)f Fm(m)806 197 y Fq(N)898 185 y Fm(>)f(K)1063 197 y Fj(1)1131 185 y Fp(suc)n(h)i(that)g(one)g(has)g Fm(f)1854 197 y Fj(2)1891 185 y Fp(\()p Fm(m)1996 197 y Fq(N)2059 185 y Fp(\))e Fk(\021)g Fm(M)h Fk(\000)20 b Fp(\()p Fm(N)30 b Fk(\000)21 b Fp(1)p Fm(=)p Fp(4\)\()p Fm(m)2888 197 y Fq(N)2970 185 y Fp(+)g(1\))3130 154 y Fj(2)3196 185 y Fm(<)29 b Fp(0.)47 286 y(Again)36 b Fm(f)339 298 y Fj(2)412 286 y Fp(is)g(a)f(p)r(olynomial)h(of)g(degree)f(t)n(w)n(o)g (in)h Fm(m)1732 298 y Fq(N)1795 286 y Fp(,)i(with)f(p)r(ositiv)n(e)f (leading)f(co)r(e\016cien)n(t)h(1)p Fm(=)p Fp(4)f(and)47 388 y(p)r(ositiv)n(e)c(discriminan)n(t,)g(so)g(that)g(there)g(exist)g (t)n(w)n(o)f(in)n(tegers)g 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b(the)g(solutions)f(whose)g(existence)h(is)f(stated)h(in) g(Lemma)f(5)h(ha)n(v)n(e)e(the)47 966 y(form)k(of)h(w)n(a)n(v)n(e)e (pac)n(k)n(ets)g(cen)n(tered)h(around)g(some)g(harmonic)g(large)f (enough.)118 1068 y(Hence)i(w)n(e)f(ha)n(v)n(e)g(pro)n(v)n(ed)f(the)i (follo)n(wing)e(result.)47 1240 y Fe(Lemma)k(7.)36 b Fn(F)-6 b(or)29 b(any)g Fm(N)37 b Fn(ther)l(e)29 b(ar)l(e)g(sets)f Fk(M)g Fn(and)h(functions)g Fm(v)2107 1252 y Fj(0)2144 1240 y Fp(\()p Fm(x;)14 b(t)p Fp(\))24 b(=)f Fm(a)p Fp(\()p Fm(!)s(t;)14 b(x)p Fp(\))i Fk(\000)f Fm(a)p Fp(\()p Fm(!)s(t;)f Fk(\000)p Fm(x)p Fp(\))p Fn(,)30 b(with)1149 1447 y Fm(a)p Fp(\()p Fm(t;)14 b(x)p Fp(\))24 b(=)1537 1369 y Fl(X)1483 1547 y Fq(m)p Fr(2M)1665 1555 y Fi(+)1725 1447 y Fm(e)1764 1413 y Fq(im)1846 1388 y Fi(2)1878 1413 y Fq(!)r(t)p Fj(+)p Fq(imx)2122 1447 y Fm(a)2166 1459 y Fq(m)2229 1447 y Fm(;)890 b Fp(\(4)p Fm(:)p Fp(11\))47 1727 y Fn(which)32 b(solve)e(the)g(Q)g(e)l(quation)g(with)g Fm(")23 b Fp(=)f(0)p Fn(.)118 1899 y 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