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the number 2701 has a curious symmetry.

Its factors are 37 and 73 (prime numbers).

Both factors are mirror reflections of each other.

In addition to this symmetry, if one adds:

2701 + 1072 (its own mirror reflection)

the sum is

3773 (same as the factors in order concatenated)

Are there any other positive integer numbers like this?

If yes, what is the next one (base 10)? (distinct factors)

If this is the only solution in base 10, what are some other solutions for other bases?

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    $\begingroup$ Have you tried generating these? Find a list of prime numbers, find the ones that are reflections of each other, multiply them and check. You might be looking for a long time, so it helps if you can program. $\endgroup$
    – Jair Taylor
    Mar 3, 2020 at 17:47
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    $\begingroup$ Do you accept answers in bases other than base 10? Questions like these are base 10-centric. $\endgroup$
    – user78090
    Mar 3, 2020 at 18:38
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    $\begingroup$ @user78090 Nice. Questions like these are very difficult to resolve theoretically so I would not expect an answer unless a second solution is found by brute force search. $\endgroup$
    – Jair Taylor
    Mar 3, 2020 at 19:06
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    $\begingroup$ The solutions mentioned are the ones I have found. Trivial in the sense of being easy to find and not contributing much to our understanding of the underlying structure (assuming it exists). I suspect there are no other solutions although I haven't checked bases above 10. For bases 2 to 10 I have checked primes up to about 200 million. $\endgroup$
    – Peter Phipps
    Mar 12, 2020 at 12:54
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    $\begingroup$ @GerryMyerson, you may have a point. Up to 10 million, in base ten, the only non-prime example is $9$. $\endgroup$
    – Peter Phipps
    Mar 13, 2020 at 15:07

1 Answer 1

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These are some results that I (my PC) have found so far, where the prime and its reverse are different.

Part 1: Bases $2$ to $10$. I have checked these for primes up to 200 million.
$52_7 \times 25_7 = 2023_7$ and $2023_7+3202_7 = 5225_7$
$37_{10} \times 73_{10} = 2701_{10}$ and $2701_{10}+1072_{10} = 3773_{10}$

Part 2: Bases $11$ to $36$ (here $A_{base}$ is decimal $10$ etc.) I have checked these for primes up to (a mere) 2 million.
$3JCFF_{22} \times FFCJ3_{22} = 2H26969A21_{22}$ and $2H26969A21_{22}+12A96962H2_{22} = 3JCFFFFCJ3_{22}$
$9J_{28} \times J9_{28} = 6J03_{28}$ and $6J03_{28}+30J6_{28} = 9JJ9_{28}$
$4CTG_{31} \times GTC4_{31} = 2CR2E202_{31}$ and $2CR2E202_{31}+202E2RC2_{31} = 4CTGGTC4_{31}$
These large solutions I found very surprising.

In addition, I found the following solutions where the prime is a single digit:
$3_4 \times 3_4 = 21_4$ and $21_4+12_4 = 33_4$
$3_7 \times 3_7 = 12_7$ and $12_7+21_7 = 33_7$
$5_6 \times 5_6 = 41_6$ and $41_6+14_6 = 55_6$
$7_8 \times 7_8 = 61_8$ and $61_8+16_8 = 77_4$
$5_{11} \times 5_{11} = 23_{11}$ and $23_{11}+32_{11} = 55_{11}$
$B_{12} \times B_{12} = A1_{12}$ and $A1_{12}+1A_{12} = BB_{12}$
$D_{14} \times D_{14} = C2_{14}$ and $C2_{14}+2C_{14} = DD_{14}$
$7_{15} \times 7_{15} = 34_{15}$ and $34_{15}+43_{15} = 77_{15}$
$H_{18} \times H_{18} = G1_{18}$ and $G1_{18}+1G_{18} = HH_{18}$
$J_{20} \times J_{20} = I1_{20}$ and $I1_{20}+1I2_{20} = JJ_{20}$
$5_{21} \times 5_{21} = 14_{21}$ and $14_{21}+41_{21} = 55_{21}$
$7_{22} \times 7_{22} = 25_{22}$ and $25_{22}+52_{22} = 77_{22}$
$B_{23} \times B_{23} = 56_{23}$ and $56_{23}+65_{23} = BB_{23}$
$N_{24} \times N_{24} = M1_{24}$ and $M1_{24}+1M_{24} = NN_{24}$
$D_{27} \times D_{27} = 67_{27}$ and $67_{27}+76_{27} = DD_{27}$
$U_{30} \times U_{30} = T1_{30}$ and $T1_{10}+1T_{30} = UU_{30}$
$W_{32} \times W_{32} = V1_{32}$ and $V1_{32}+1V_{32} = WW_{32}$
$H_{35} \times H_{35} = 89_{35}$ and $89_{35}+98_{35} = HH_{35}$

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