Yeah but they usually require RSA to be used in some rather unusual and bad way.
For example, encrypting one message with many different public keys can be broken with chinese remainder theorem and Nth roots. This reveals the message without factoring any key. This is why randomized padding (among other things) is a must with RSA.
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The RSA problem in article doesn't mention that RSA's difficulty is based on MODULUS prime factorization, not simple prime factorization.
https://en.wikipedia.org/wiki/RSA_(cryptosystem)
Are there any known attack methods that don't involve factoring the public key into two primes?
Yeah but they usually require RSA to be used in some rather unusual and bad way.
For example, encrypting one message with many different public keys can be broken with chinese remainder theorem and Nth roots. This reveals the message without factoring any key. This is why randomized padding (among other things) is a must with RSA.
There's a footnote along these lines that links to the actual algorithm.