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Zhi-Wei Sun, <a href="https://doi.org/10.1016/j.jnt.2013.02.003">On functions taking only prime values</a>, J. Number Theory , 133 (2013), 2794-2812.

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Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), 2794-2812.

Zhi-Wei Sun, <a href="http://arxiv.org/abs/1304.5988">On primes in arithmetic progressionsThe least modulus for which consecutive polynomial values are distinct</a>, arXiv:1304.5988 [math.NT], 2013-2015.

Zhi-Wei Sun, <a href="https://doi.org/10.1016/j.jnt.2013.02.003">On functions taking only prime values</a>, J. Number Theory 133(2013), 2794-2812.

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Zhi-Wei Sun also conjectured that for each n>17 the least Fibonacci number modulo which 1^2,2^2,...,n^2 are pairwise incongruent is just the first Fibonacci prime greater than 2n-1.

This phenomenon might happen for some other Lucas sequences u_0,u_1,... given by u_0 = 0, u_1 = 1, and u_{k+1} = Au_A*u_k-Bu_B*u_{k-1} for k>0, with A>0 and B (nonzero) relatively prime and A^2 > 4B.

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Zhi-Wei Sun also made conjectured that for n>17 the least Fibonacci number modulo which 1^2,2^2,...,n^2 are pairwise incongruent is just the following general conjecture:first Fibonacci prime greater than 2n-1.

Let A>0 and B (nonzero) be relatively prime integers with A^2>4B. Define u_0=0, u_1=1, and u_{k+1}=Au_k-Bu_{k-1} for k>0. If n is large enough, then the least integer m>1 such that 1^2,2^2,...,n^2 are pairwise incongruent modulo u_m is just the first prime p with u_p the least prime in the Lucas sequence u_0,u_1,u_2,... greater than 2n-1.

For example, This phenomenon might happen for n>17 the least Fibonacci number modulo which some other Lucas sequences u_0,u_1^2,2^2,,...,n^2 are pairwise incongruent is just the first Fibonacci given by u_0=0, u_1=1, and u_{k+1} = Au_k-Bu_{k-1} for k>0, with A>0 and B (nonzero) relatively prime greater than 2n-1and A^2 > 4B.

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