## Prime arithmetic progression with 3 as a first term

If   $a$ and   $b$   are the first two terms of an arithmetic progression,
then the third term is   $2 \,b \; - \; a$

$a, \; b, \; c$

$b \; - \; a \; = \; c \; - \; b$
$c \; = \; 2 \,b \; - \; a$

Every prime > 3   has one of the forms   $6 \,k -l$   or   $6 \,k + 1$

If   $6 \, m - 1$   and   $6 \, n + 1$   are the first two terms of an arithmetic progression,
then the third term is the composite :

$c \; = \; 2 \,(6 \, n + 1) \; - \; (6 \, m - 1) \; = \; 3 (-2 \, m + 4 \, n + 1)$

If the first two terms of an arithmetic progression are   $6 \, p + 1$   and   $6 \, q - 1$,
then the third term is the composite

$c \; = \; 2 \,(6 \, q - 1) \; - \; (6 \, p + 1) \; = \; 3 \,(-2 \, p + 4 \, q - 1)$

It follows that any prime arithmetic progression with 3 as a first term cannot have more than three terms.

Here are examples where the second term has   1,2,…, 7   digits:

3,   5,   7
3,   11,   19
3,   101,   199
3,   1021,   2039
3,   10007,   20011
3,   100003,   200003
3,   5003261,   10006519

and the common difference is respectively:
2,   8,   98,   1018,   10004,   100000,   5003258

Find other sequences where the second term is larger than 7 digits.

Paul:

{3,   10000223,   20000443,   10000220}
{3,   100000123,   200000243,   100000120}
{3,   1000000007,   2000000011,   1000000004}
{3,   1000000014,   20000000291,   10000000144}

K. D. Bajpai:

{3,   100000000237,   200000000471,   100000000234}
{3,   1000000000063,   2000000000123,   1000000000060}
{3,   10000000000643,   20000000001283,   10000000000640}
{3,   100000000000487,   200000000000971,   100000000000484}
{3,   1000000000000091,   2000000000000179,   1000000000000088}
{3,   10000000000000613,   20000000000001223,   10000000000000610 }

In the format {3,   p,   q,   common difference}

math grad - Interest: Number theory
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### 8 Responses to Prime arithmetic progression with 3 as a first term

1. paul says:

Here are a few more, Format {p1,p2,p3,diff}

{3,10000223,20000443,10000220}
{3,100000123,200000243,100000120}
{3,1000000007,2000000011,1000000004}
{3,10000000147,20000000291,10000000144}

Paul.

2. K. D. Bajpai says:

Some more, in the same Format { p1, p2, p3, diff}
{ 3, 100000000237, 200000000471, 100000000234 }
{ 3, 1000000000063, 2000000000123, 1000000000060 }
{ 3, 10000000000643, 20000000001283, 10000000000640 }
{ 3, 100000000000487, 200000000000971, 100000000000484 }
{ 3, 1000000000000091, 2000000000000179, 1000000000000088 }
{ 3, 10000000000000613, 20000000000001223, 10000000000000610 }

3. K.D. BAJPAI says:

In all the examples where the second term has 1,2,…,17 digits, the (smallest) first set of {3, p, q } is considered, except for two digits where the fourth set i.e.{3, 23, 43 } with common difference 20 is considered.
It is suggested that the (smallest) first set {3, 11, 19 } with common difference 8 for two digits may be considered to maintain the symmetry.

• benvitalis says:

You’re right.

• benvitalis says:

I made the change on the blog. Thanks for pointing that out.

4. K.D. BAJPAI says:

I think the 3rd line { 3, 23, 43 } is now not required.