Problem A prime\left trio is a collection of three prime numbers \{p, q, r\} in arithmetic progression, with common difference q-p=r-q. For example, the prime trio \{3, 5, 7\} h…

Question

Problem

A
`prime\left trio` is a collection of three prime numbers
`\{p, q, r\}` in arithmetic progression, with common difference
`q-p=r-q`. For example, the prime trio
`\{3, 5, 7\}` has common difference 2.
Prove that there is no prime trio with common difference 70.
P.S. I want actual proof.

Answer 1

Proof below

Explanation:

General outline

1. Lemma regarding remainders of non-small primes divided by 6
2. Check
   `p=2` and
   `p=3` directly
3. Proof the rest by contradiction

Lemma

If
`p` is prime such that
`p \\eq2` and
`p \\eq 3`, then
`p` divided by 6 has remainder of 1 or 5.

Case 1: p/6 has remainder 0, 2, or 4.

Then there exists some natural number
`n` such that either
`p=6n`,
`p=6n+2`, or
`p=6n+4`. However,

• if
  `p=6n`, then
  `p=2*3n`,
• if
  `p=6n+2`, then
  `p=2*(3n+1)`, and
• if
  `p=6n+4`, then
  `p=2*(3n+2)`.

By definition of divisibility,
`p` is divisible by 2. Since
`p \\eq2`,
`p` cannot be prime. This contradiction implies
`p` cannot have a remainder of 0, 2, or 4 when divided by 6.

Case 2: p/6 has remainder 3.

Then there exists some natural number
`n` such that
`p=6n+3`. However, if
`p=6n+3`, then
`p=3*(2n+1)`, so
`p` is divisible by 3. Since
`p \\eq 3`,
`p` is not prime. This contradiction implies
`p` cannot have a remainder of 3 when divided by 6.

Therefore, if
`p` is a prime number such that
`p \\eq2` and
`p \\eq 3`, then when divided by 6,
`p` must have a remainder of 1 or 5.

Main proof of no prime trio with common difference of 70.

Check p=2

Consider
`p=2`. Then
`q=2+70=72`. However,
`72=2*36`, so
`q` is not prime. Hence,
`p \\eq2`.

Check p=3

Consider
`p=3`. Then
`q=3+70=73`, and
`r=73+70=143`. However,
`143=11*13`, so
`r` is not prime. Hence,
`p \\eq 3`.

So, thus far, we've proven that if there is a prime trio with a common difference of 70, that
`p \\eq2` and
`p \\eq 3`.

Proof of the rest of primes by contradiction

By way of contradiction, assume that there does exist some prime trio
`\{ p,q,r \}` such that
`q-p=r-q=70`, and
`p \\eq2` and
`p \\eq 3`.

Then, by the Lemma proven earlier, when
`p` is divided by 6, it must have a remainder of 1 or 5.

Case 1: p/6 has remainder 5

Assume that
`p` has remainder 5 when divided by 6. Then, there exists some natural number
`n`, such that
`p=6n+5`.

`q=p+70\\q=(6n+5)+70\\q=6n+75\\q=3*(2n+25)`

Since
`n` was a natural number, and the natural numbers are closed over multiplication and addition (meaning, multiplying and adding more natural numbers results in another natural number), then
`q` is divisible by 3.

Since
`p` is prime,
`0 < p`, which implies
`0+70 < p+70`

Since
`q=p+70`,
`70 < q`, so
`q` cannot equal 3. Since
`q` is divisible by 3, but
`q \\eq 3`,
`q` is not prime, which is a contradiction to the existence of this prime trio.

Therefore, either
`p` cannot have a remainder of 5 when divided by 6, or the prime trio does not exist.

Case 2: p/6 has remainder 1

Assume that
`p` has remainder 1 when divided by 6. Then, there exists some natural number
`n`, such that
`p=6n+1`.

`q=p+70\\q=(6n+1)+70\\r=q+70\\r=((6n+1)+70)+70\\r=6n+141\\r=3*(2n+47)`

Since
`n` was a natural number, and the natural numbers are closed over multiplication and addition (meaning, multiplying and adding more natural numbers results in another natural number),
`r` is divisible by 3.

Since
`q` is prime,
`0 < q`, implying
`0 +70 < q+70`.

Since
`r=q+70`, then
`70 < r`. Therefore,
`r\\eq 3`.

Since
`r` is divisible by 3, but
`r\\eq 3`,
`r` is not prime, which is a contradiction to the existence of this prime trio (with a common difference of 70). Therefore, either
`p` cannot have a remainder of 1 when divided by 6, or the prime trio does not exist.

Since
`p` was prime, by our lemma
`p` must have either a remainder of 1 or 5. Since both remainder possibilities resulted in a contradiction, our contradiction assumption is false, so there cannot exist a prime trio such that their common difference is 70.