A Poulet-number is a composite number $N$ satisfying $$2^{N-1}\equiv 1\mod N$$ A palindrome is a positive integer with a digit string in base $10$ which remains the same if it is written down backwards.
If $3p$ is a Poulet-number and $p$ a palindrome , it seems that $p$ must begin (and therefore end) with digit $7$. The examples upto $p=10^{18}$ are :
23001 7667 [11, 1; 17, 1; 41, 1]
212421 70807 [11, 1; 41, 1; 157, 1]
23054601 7684867 [17, 1; 251, 1; 1801, 1]
235201461 78400487 [11, 1; 41, 1; 131, 1; 1327, 1]
Why is $p$ always of the form $7...7$ ?
Why is $p$ divisible by $41$ , if it is divisible by $11$ ?
Any counter-example?