The secret code, the math enigma of “Le Monde” n° 56

The bus line I take every day serves the city's math lab, and as a result, I regularly overhear strange conversations. Just yesterday, I overheard (in spite of myself) an exchange between the two people sitting behind me.

— Do you have the new access code to the conference room? I heard they changed it yesterday?

—Yes, I received it, replied his colleague, besides, I noticed some amusing details about these six digits! The number formed by the first and second digits is a prime number. The number formed by the second and third digits is a square number. The number formed by the third and fourth digits is again prime, and the number formed by the fourth and fifth is again a square. As for the number formed by the fifth and sixth, it is…

At that moment, a motorcycle noisily passed by the bus, preventing me from hearing the end of the sentence. A few seconds later, the first mathematician continued hesitantly.

— I don't think these clues will allow me to find the code...

— Ah yes, continued the mathematician, the sixth digit is not equal to the fourth!

After a few more seconds, I saw the man's face light up.

— Perfect, he exclaimed. This time I've got it!

For my part, I really didn't see how I could have found the code when one of the clues had escaped me. Unless...

Could you figure out the new six-digit code for the conference room?

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