Reply To: Maths

[Kingswinford-Warriors-Alumni]

[General message: Sorry for the abundance of posts recently!]

#98720 (Hints for #98066)

Let’s try again! I have reverted to a ROT13 cipher (i.e. Caesar shift of 13), combined with a ROT5 (i.e. Caesar shift of 5 on the digits only) to scramble the numbers slightly. Note that this does not change things like symbols or brackets, so some of the structure of the hints has been retained.

You view these hints at your own risk. (Disclaimer)

Everything fine? Great. On with the hints!

HINT 0 (solutions)

Gur bayl inyhr bs (n,o,c) gung jbexf vf (6,6,7).

HINT 1 (FLT)

Srezng'f Yvggyr Gurberz fgngrf gung, sbe nal cevzr c naq vagrtre x pbcevzr gb c, x^(c-6) unf erznvaqre 6 ba qvivqvat ol c. (Be x^(c-6)=6 zbq c.) Zber hfrshyyl sbe guvf ceboyrz, vg nyfb zrnaf gung sbe nal cevzr c naq NAL vagrtre x, x^c naq x unir gur fnzr erznvaqre ba qvivqvat ol c. Guvf zrnaf gung jr znl ercynpr, fnl, n^c ol n va gur ceboyrz naq fgvyy cerfreir gur erznvaqre ba qvivqvat ol c...

HINT 2 (first steps)

Nf uvagrq ng va Uvag 6, hfr SYG gb zbqvsl gur yrsg-unaq fvqr bs gur rkcerffvba, juvyfg cerfreivat vgf erznvaqre ba qvivqvat ol c. (Gung vf, pbafvqre obgu fvqrf zbq c.)
Qrqhpr gung n+o=5 (zbq c), be gung c qvivqrf n+o.

HINT 3 (bounds on a,b,a+b)

Uvag 7 fgngrq gung c qvivqrf n+o.
Vs c=7 gura jr unir n^7+o^7=7, juvpu vf rnfl gb purpx, fb nffhzr c>7. Abgr gung c vf bqq. Yrg a=(c+6)/7.

Jevgr a^c naq c! nf sbyybjf (ivrj va n zbabfcnprq sbag):

a^c = [a*a] * [a* a   ] * ... * [ a   * a   ] * a
c!  = [6*c] * [7*(c-6)] * ... * [(a-6)*(a+6)] * a

Cebir gung rnpu bs gur oenpxrgf sbe a^c ner ynetre guna gubfr sbe c!. Urapr, fvapr n^c < c! < a^c, qrqhpr gung n<=(a-6). Fvzvyneyl fubj o<=(a-6), naq urapr fubj gung n+o<c.

HINT 4 (final steps)

(V nz nffhzvat c>7 urer, nf fgngrq va Uvag 8...)
Uvag 7 fnvq gung c qvivqrf n+o.
Uvag 8 fnvq gung n+o<c.
n naq o ner cbfvgvir, fb jr nyfb unir n+o>5.
Qrevir n pbagenqvpgvba.