# Smallest 17 Primes in Arithmetic Progression

## Theorem

The smallest $17$ primes in arithmetic progression 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 progression of primes does not extend to $n < 0$.

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

## Historical Note

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

It was discovered in $1977$ by Sol Weintraub.

Since that time, plenty longer have been found.