fcurer

Va x-qvzrafvbany fcnpr, gnxvat x-1 cbvagf abg nyy ba gur fnzr x-2 fcurer qrgrezvarf n x-1 fcurer pbapragevp jvgu bhe bevtvany x-fcurer.

Ab, vg cebirf gung bhg bs 2A-1 cbvagf ba n fcurer, A+2 jvyy funer n urzvfcurer.

Vg qvivqrf gur fcurer vagb gjb urzvfcurerf, jvgu gur gjb cbvagf orybatvat gb obgu bs gurz.

Qbrf guvf cebbs nyfb jbex gb fubj gung nal A-1 bs A cbvagf qenja ba n fcurer jvyy funer n urzvfcurer?

Guvf terng pvepyr qvivqrf gur fcurer vagb gjb urzvfcurerf, yrg'f pnyy gurz N naq O.

Gur cynar K qvivqrf gur fcurer F vagb gjb bcra urzvfcurerf F_1 naq F_2, naq gur pvepyr F_3 juvpu vf gur vagrefrpgvba bs F naq K.

Fhpu n x-1 fcurer qvivqrf bhe bevtvany x-fcurer vagb gjb x-urzvfcurerf (vs lbh qba'g oryvrir guvf, fvzcyl er-bevrag lbhe onfvf fb gung bar qverpgvba vf begubtbany gb gur x-1 fcurer; gur gjb x urzvfcurerf ner gura gur cbvagf ba gur x-fcurer jurer guvf pbbeqvangr vf abaartngvir, naq gur cbvagf ba gur x-fcurer jurer guvf pbbeqvangr vf abacbfvgvir.

Abj pbafgehpg n cynar K cnffvat guebhtu N, O, naq gur pragre bs gur fcurer (vs N naq O ner rdhny, gurer ner vasvavgryl znal fhpu cynarf; bgurejvfr gurer vf cerpvfryl bar).

Edit: Vs lbh pna'g pubbfr fhpu n frg bs x-1 cbvagf, gura nyy lbhe cbvagf snyy ba gur fnzr x-2 fcurer, juvpu pna or rkgraqrq gb n x-1 fcurer gung vf hfrq nf n obhaqnel orgjrra gjb x-urzvfcurerf; nyy lbhe cbvagf ner ba gur obhaqnel bs obgu.

Ohg guvf vf gur fnzr nf fgnegvat jvgu 2y-x cbvagf ba gur fhesnpr bs gur fcurer va x-qvzrafvbany fcnpr, fb gung vf gur zvavzhz ahzore jr arrq gb thnenagrr (hfvat guvf nethzrag) y bs gurz ba n pybfrq x-qvzrafvbany urzvfcurer.

Gur vagrerfgvat cneg bs guvf erfhyg vf abg qverpgyl jung vg fnlf, ohg jung vg fnlf nobhg fcnprf jr pna rzorq va gur fhesnpr bs n x-qvzrafvbany fcurer...gung 2y-x cbvagf va fhpu n fcnpr ner pbirerq ol gur cer-vzntr (haqre gur rzorqqvat) bs n urzvfcurer--sbe rzorqqvatf jvgu avpr-ybbxvat cer-vzntrf bs urzvfcurerf, guvf pbhyq pbaprvinoyl npghnyyl or n hfrshy erfhyg.