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关于Erdos的一个猜想
Introduction:
PaulErdőswasaprolificmathematicianwhomadesignificantcontributionstovariousbranchesofmathematics,includingnumbertheory,graphtheory,andcombinatorics,amongothers.HewasknownforhiscollaborativeapproachtomathematicsandtheErdősnumber,ameasureofamathematician'scollaborationdistancefromErdős,whichhasbecomeafamousconceptinthemathematicscommunity.OneofErdős'smostfamousconjecturesistheErdősdiscrepancyproblem,whichremainsunsolved.
ErdősDiscrepancyProblem:
TheErdősdiscrepancyproblemisaconjecturethatremainsunsolvedsinceitwasproposedbyPaulErdősinthe1930s.Theconjectureconcernsasequenceofsigns{+1,-1}andseekstoprovethatthediscrepancyofthissequenceisunbounded.Thediscrepancyofasequenceisdefinedasthemaximumabsolutedifferencebetweenthenumberof+1'sand-1'sinanycontiguoussubsequenceofthesequence.Inotherwords,ifwehaveasequence{a1,a2,…,an},thediscrepancyisthemaximumvalueof|(a1+a2+...+aj)-(a(j+1)+a(j+2)+...+an)|,where1<=j<n.
TheErdősdiscrepancyproblemaimstoprovethatforanyinfinitesequence{a1,a2,…},whereai=+1or-1,thereexistsaninfinitesubsequence{aj1,aj2,…}suchthatthediscrepancyofthesubsequenceisgreaterthananyarbitrarilylargepositiveinteger.Thatis,nomatterhowlargewechooseapositiveintegerK,wecanalwaysfindasubsequencewhosediscrepancyislargerthanK.
Thisconjecturehasfar-reachingimplicationsforvariousareasofmathematics,includingnumbertheory,probabilitytheory,andcombinatorics,asitrelatestothedistributionandrandomnessofsequences.Asaresult,theErdősdiscrepancyproblemhasbecomeanactiveareaofresearchinmathematics,andmanymathematicianshavetriedtosolveitovertheyears.
AttemptstoSolvetheErdősDiscrepancyProblem:
SeveralmathematicianshaveattemptedtosolvetheErdősdiscrepancyproblemovertheyears,buttodate,noonehasbeenabletoproveordisprovetheconjecture.Theproblemremainsoneofthemostfascinatingandchallengingopenproblemsinmathematics.
Oneapproachtosolvingtheprobleminvolvesusingprobabilistictechniquestoshowthatasequenceofsignscannothavesmalldiscrepancyincertaincases.Forexample,in1975,H.N.Shapiroan