Problem 1477

To prove that at least 4 of the prime numbers $p_1, p_2, \dots, p_{2021}$ are less than 2021, given that the sum $p_1^4 + p_2^4 + \dots + p_{2021}^4$ is divisible by 6060, we will consider the prime factorization of 6060 and analyze the sum modulo these prime factors.

1. Prime Factorization of 6060:
$$6060 = 2^2 \times 3 \times 5 \times 101$$
We need to ensure that the sum $p_1^4 + p_2^4 + \dots + p_{2021}^4$ is divisible by each of these prime factors.

2. Modulo 2:
$$6060 \equiv 0 \pmod{2}$$
Since $p_1^4 + p_2^4 + \dots + p_{2021}^4$ must be even, at least one of the primes $p_i$ must be 2. If none of the primes were 2, the sum would be odd, which contradicts the requirement that it is divisible by 2.

3. Modulo 3:
$$6060 \equiv 0 \pmod{3}$$
For any integer $x$, $x^4 \equiv 0$ or $1 \pmod{3}$. If none of the primes were 3, each $p_i^4 \equiv 1 \pmod{3}$. Thus, the sum would be:
$$p_1^4 + p_2^4 + \dots + p_{2021}^4 \equiv 2021 \equiv 2 \pmod{3}$$
This is a contradiction since the sum must be 0 modulo 3. Therefore, we need at least two primes to be 3 to make the sum 0 modulo 3.

4. Modulo 5:
$$6060 \equiv 0 \pmod{5}$$
For any integer $x$, $x^4 \equiv 0$ or $1 \pmod{5}$. If none of the primes were 5, each $p_i^4 \equiv 1 \pmod{5}$. Thus, the sum would be:
$$p_1^4 + p_2^4 + \dots + p_{2021}^4 \equiv 2021 \equiv 1 \pmod{5}$$
This is a contradiction since the sum must be 0 modulo 5. Therefore, we need at least one prime to be 5 to make the sum 0 modulo 5.

5. Modulo 101:
$$6060 \equiv 0 \pmod{101}$$
For any integer $x$, $x^4 \equiv 0$ or $1 \pmod{101}$. If none of the primes were 101, each $p_i^4 \equiv 1 \pmod{101}$. Thus, the sum would be:
$$p_1^4 + p_2^4 + \dots + p_{2021}^4 \equiv 2021 \equiv 1 \pmod{101}$$
This is a contradiction since the sum must be 0 modulo 101. Therefore, we need at least one prime to be 101 to make the sum 0 modulo 101.

From the above analysis, we conclude that we must have the primes $2, 3, 3, 5$ among the 2021 primes to satisfy the divisibility conditions. All these primes are less than 2021.

$\blacksquare$