2
$\begingroup$

Surely I'm not the first to observe! But it's a bit of a cheat: In which base do we read the bases when stating $a_b=b_a$? (I write $a_b$ for $a$ base $b$.)

Suggestion: Like in the title, in a "master" base $B=10$ (for the start - feel free to generalize). Also: small variables shall be numbers, large digits.

With a Python program, I found:

  • $a_a=a_a$ (duh)
  • $10_x = x_{10}$ (any $x$, only solution with different lengths)
  • $1X_{1Y} = 1Y_{1X}$ (any $X,Y$)
  • $XY_{UV} = UV_{XY}$ where $X-Y=U-V=1$

More length $2$ solutions (have some pattern, could be compressed a bit):

34 22 70 36 23 75 38 24 80 46 22 94 46 34 142 49 23 101 49 36 153 52 31 157 56 33 171 58 22 118 58 34 178 58 46 238 72 41 289 73 31 220 73 52 367 74 42 298 78 44 316 79 33 240 79 56 401 92 51 461 94 31 283 94 52 472 94 73 661 96 53 483

Note that length $2$ solutions $ij,kl$ are base $B$-independent: $j(k-1)=l(i-1)$. This gives the whole list above.

Both length $3$ (biquadratic in $B$, the resultant has a small-ish factor which gives "accidental" solutions):

150 210 45150 164 230 54284 178 250 64258

Solutions with one length$>3$ seem to be absent for elliptic curve magic. (For time reasons, I tested only length$=4$ and $B\leq 11$.)

Can you prove these are all solutions? (Partial results welcome, this is heavy number theory.)

$\endgroup$
4
  • 4
    $\begingroup$ Note you are expressing the $b$ in "$b_a$" as a digits assumed to be in base $10$. If you express them in digits assumed to be in base you always get the result. $\endgroup$ Commented Nov 18 at 18:38
  • 3
    $\begingroup$ My favourite way of indicating the base I ever saw was in a book that would right things like $11_\text{ten}=13_\text{eight}$. I'm not convinced numeral representations of the base are an improvement. $\endgroup$ Commented Nov 19 at 12:31
  • $\begingroup$ Isn't $(9000, 300)$ solution base 10? $9000_{300} = 9\cdot 300^3 = 3^5 \cdot 10^6$, and $300_{9000} = 3\cdot 9000^2 = 3^5 \cdot 10^6$? $\endgroup$ Commented Nov 19 at 20:33
  • $\begingroup$ @mihaild: Good catch. Should generalize heavily. (At least $4000_{200}$, but possibly 80....20... also works?!) $\endgroup$ Commented Nov 22 at 19:58

0

Reset to default

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.