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?
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 $89_{35}+98_{35} = HH_{35}$