## Task 13/63

A mathematician had his bicycle stolen. When asked for his bicycle number, he replied: "You can calculate the number from the following information:

a) If you add the square of the first digit to the square of the second digit, you get the square of the third digit.

b) If you subtract the second digit from the first digit, you get the fifth digit increased by 1.

c) The second digit is equal to the fourth, the third digit is equal to the sixth and to the seventh."

What was the mathematician's bicycle number?
The number is provisionally denoted as sbcdef $g$, where each character represents an integer between 0 and 9 (inclusive), and the notation follows the decimal positional system.

After c) it can be simplified: $b=d, c=f=g$ so the number becomes abcbecc. According to a) $a^{2}+b^{2}=c^{2}$. Therefore, $a, b$, and $c$ are Pythagorean numbers; the only Pythagorean triplet that satisfies the condition that all numbers are less than 10 is the triplet $3 ; 4 ; 5$. This narrows down the number. It is either 1. $3454 e 55$ or 2. $5353 e 55$.

Now, calculate $e$ according to b): $a-b=e+1$ results in $e=a-b-1$.

1.) With $a=3$ and $b=4$, we get $e=-2$. A negative number is clearly not feasible, so option 1 is ruled out.

2.) With $a=4$ and $b=3$, we get $e=0$. Therefore, the complete number is 4353055.