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An earlier MO questionAn earlier MO question highlighted Giuga's Conjecture:

A positive integer $n>1$ is prime if and only if $$\sum_{k=1}^{n-1} k^{n-1} \equiv -1 \pmod{n}$$

For example, for the prime $n=5$, the sum is $$ 1^4 + 2^4 + 3^4 + 4^4 = 1+16+81+256 \equiv 1+1+1+1\,(\rm{mod}\, 5)\equiv -1(\rm{mod}\, 5) \;. $$ My question is: Is this conjecture in some sense central in the pursuit of prime properties (as is—surely—e.g., Yitang Zhang's and James Maynard's advances related to prime pairs), or is it rather perceived as peripheral to the general understanding of primes? I ask not only because of the spectacular recent advances, but because the iff nature of the conjecture seems almost too good to be true, even though it has been verified out to $10^{13800}$.

An earlier MO question highlighted Giuga's Conjecture:

A positive integer $n>1$ is prime if and only if $$\sum_{k=1}^{n-1} k^{n-1} \equiv -1 \pmod{n}$$

For example, for the prime $n=5$, the sum is $$ 1^4 + 2^4 + 3^4 + 4^4 = 1+16+81+256 \equiv 1+1+1+1\,(\rm{mod}\, 5)\equiv -1(\rm{mod}\, 5) \;. $$ My question is: Is this conjecture in some sense central in the pursuit of prime properties (as is—surely—e.g., Yitang Zhang's and James Maynard's advances related to prime pairs), or is it rather perceived as peripheral to the general understanding of primes? I ask not only because of the spectacular recent advances, but because the iff nature of the conjecture seems almost too good to be true, even though it has been verified out to $10^{13800}$.

An earlier MO question highlighted Giuga's Conjecture:

A positive integer $n>1$ is prime if and only if $$\sum_{k=1}^{n-1} k^{n-1} \equiv -1 \pmod{n}$$

For example, for the prime $n=5$, the sum is $$ 1^4 + 2^4 + 3^4 + 4^4 = 1+16+81+256 \equiv 1+1+1+1\,(\rm{mod}\, 5)\equiv -1(\rm{mod}\, 5) \;. $$ My question is: Is this conjecture in some sense central in the pursuit of prime properties (as is—surely—e.g., Yitang Zhang's and James Maynard's advances related to prime pairs), or is it rather perceived as peripheral to the general understanding of primes? I ask not only because of the spectacular recent advances, but because the iff nature of the conjecture seems almost too good to be true, even though it has been verified out to $10^{13800}$.

It's Maynard, not Maynar
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Stanley Yao Xiao
Stanley Yao Xiao
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An earlier MO question highlighted Giuga's Conjecture:

A positive integer $n>1$ is prime if and only if $$\sum_{k=1}^{n-1} k^{n-1} \equiv -1 \pmod{n}$$

For example, for the prime $n=5$, the sum is $$ 1^4 + 2^4 + 3^4 + 4^4 = 1+16+81+256 \equiv 1+1+1+1\,(\rm{mod}\, 5)\equiv -1(\rm{mod}\, 5) \;. $$ My question is: Is this conjecture in some sense central in the pursuit of prime properties (as is—surely—e.g., Yitang Zhang's and James Maynar'sMaynard's advances related to prime pairs), or is it rather perceived as peripheral to the general understanding of primes? I ask not only because of the spectacular recent advances, but because the iff nature of the conjecture seems almost too good to be true, even though it has been verified out to $10^{13800}$.

An earlier MO question highlighted Giuga's Conjecture:

A positive integer $n>1$ is prime if and only if $$\sum_{k=1}^{n-1} k^{n-1} \equiv -1 \pmod{n}$$

For example, for the prime $n=5$, the sum is $$ 1^4 + 2^4 + 3^4 + 4^4 = 1+16+81+256 \equiv 1+1+1+1\,(\rm{mod}\, 5)\equiv -1(\rm{mod}\, 5) \;. $$ My question is: Is this conjecture in some sense central in the pursuit of prime properties (as is—surely—e.g., Yitang Zhang's and James Maynar's advances related to prime pairs), or is it rather perceived as peripheral to the general understanding of primes? I ask not only because of the spectacular recent advances, but because the iff nature of the conjecture seems almost too good to be true, even though it has been verified out to $10^{13800}$.

An earlier MO question highlighted Giuga's Conjecture:

A positive integer $n>1$ is prime if and only if $$\sum_{k=1}^{n-1} k^{n-1} \equiv -1 \pmod{n}$$

For example, for the prime $n=5$, the sum is $$ 1^4 + 2^4 + 3^4 + 4^4 = 1+16+81+256 \equiv 1+1+1+1\,(\rm{mod}\, 5)\equiv -1(\rm{mod}\, 5) \;. $$ My question is: Is this conjecture in some sense central in the pursuit of prime properties (as is—surely—e.g., Yitang Zhang's and James Maynard's advances related to prime pairs), or is it rather perceived as peripheral to the general understanding of primes? I ask not only because of the spectacular recent advances, but because the iff nature of the conjecture seems almost too good to be true, even though it has been verified out to $10^{13800}$.

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Joseph O'Rourke
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Giuga's Conjecture: Central or Peripheral?

An earlier MO question highlighted Giuga's Conjecture:

A positive integer $n>1$ is prime if and only if $$\sum_{k=1}^{n-1} k^{n-1} \equiv -1 \pmod{n}$$

For example, for the prime $n=5$, the sum is $$ 1^4 + 2^4 + 3^4 + 4^4 = 1+16+81+256 \equiv 1+1+1+1\,(\rm{mod}\, 5)\equiv -1(\rm{mod}\, 5) \;. $$ My question is: Is this conjecture in some sense central in the pursuit of prime properties (as is—surely—e.g., Yitang Zhang's and James Maynar's advances related to prime pairs), or is it rather perceived as peripheral to the general understanding of primes? I ask not only because of the spectacular recent advances, but because the iff nature of the conjecture seems almost too good to be true, even though it has been verified out to $10^{13800}$.