Why does some key pair for RSA return same value

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Certain RSA key pairs, such as (5,7) and (13,19), exhibit a unique behavior where they yield the same result when encrypting and decrypting. This raises the question of whether there is a mathematical pattern that explains this phenomenon. The discussion highlights the importance of clarifying the conventions used in RSA, particularly the encoding exponent and modulus. One participant points out a potential misunderstanding regarding modular arithmetic, suggesting that specific examples would help clarify the inquiry. Overall, the conversation centers on the mathematical principles underlying RSA encryption and the need for precise definitions to explore this behavior further.
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There are certain key pairs of RSA that will return the same exact value when encrypt and decrypt, for example (5,7) and (13,19)

is there a mathematical pattern that describes this behavior?

Any advice would be appreciated
 
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What convention are you using? The only one I've seen is (encoding exponent, modulus), but 2^5 \equiv 4 \pmod 7 and 2^7 \equiv 3 \pmod 5, so I presume I have misunderstood your question. Can you give a more specific example?
 
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