Smallest 17 Primes in Arithmetic Sequence

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Theorem

The smallest $17$ primes in arithmetic sequence are:

$3\,430\,751\,869 + 87\,297\,210 n$

for $n = 0, 1, \ldots, 16$.


Proof

First we note that:

$3\,430\,751\,869 - 87\,297\,210 = 3\,343\,454\,659 = 17\,203 \times 194\,353$

and so this arithmetic sequence of primes does not extend to $n < 0$.


\(\ds 3\,430\,751\,869 + 0 \times 87\,297\,210\) \(=\) \(\ds 3\,430\,751\,869\) which is prime
\(\ds 3\,430\,751\,869 + 1 \times 87\,297\,210\) \(=\) \(\ds 3\,518\,049\,079\) which is prime
\(\ds 3\,430\,751\,869 + 2 \times 87\,297\,210\) \(=\) \(\ds 3\,605\,346\,289\) which is prime
\(\ds 3\,430\,751\,869 + 3 \times 87\,297\,210\) \(=\) \(\ds 3\,692\,643\,499\) which is prime
\(\ds 3\,430\,751\,869 + 4 \times 87\,297\,210\) \(=\) \(\ds 3\,779\,940\,709\) which is prime
\(\ds 3\,430\,751\,869 + 5 \times 87\,297\,210\) \(=\) \(\ds 3\,867\,237\,919\) which is prime
\(\ds 3\,430\,751\,869 + 6 \times 87\,297\,210\) \(=\) \(\ds 3\,954\,535\,129\) which is prime
\(\ds 3\,430\,751\,869 + 7 \times 87\,297\,210\) \(=\) \(\ds 4\,041\,832\,339\) which is prime
\(\ds 3\,430\,751\,869 + 8 \times 87\,297\,210\) \(=\) \(\ds 4\,129\,129\,549\) which is prime
\(\ds 3\,430\,751\,869 + 9 \times 87\,297\,210\) \(=\) \(\ds 4\,216\,426\,759\) which is prime
\(\ds 3\,430\,751\,869 + 10 \times 87\,297\,210\) \(=\) \(\ds 4\,303\,723\,969\) which is prime
\(\ds 3\,430\,751\,869 + 11 \times 87\,297\,210\) \(=\) \(\ds 4\,391\,021\,179\) which is prime
\(\ds 3\,430\,751\,869 + 12 \times 87\,297\,210\) \(=\) \(\ds 4\,478\,318\,389\) which is prime
\(\ds 3\,430\,751\,869 + 13 \times 87\,297\,210\) \(=\) \(\ds 4\,565\,615\,599\) which is prime
\(\ds 3\,430\,751\,869 + 14 \times 87\,297\,210\) \(=\) \(\ds 4\,652\,912\,809\) which is prime
\(\ds 3\,430\,751\,869 + 15 \times 87\,297\,210\) \(=\) \(\ds 4\,740\,210\,019\) which is prime
\(\ds 3\,430\,751\,869 + 16 \times 87\,297\,210\) \(=\) \(\ds 4\,827\,507\,229\) which is prime


But note that $3\,430\,751\,869 + 17 \times 87\,297\,210 = 4\,914\,804\,439 = 41 \times 97 \times 1 235807$ and so is not prime.


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In particular: It remains to be shown that there are no smaller such APs
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Historical Note

David Wells, in his Curious and Interesting Numbers of $1986$ reported that this was the $2$nd longest known arithmetic sequence of prime numbers.

It was discovered in $1977$ by Sol Weintraub.

Since that time, plenty longer have been found.


Sources

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