are prime numbers such that the numbers

, ,

form an arithmetic progression :

Find other such numbers

The problem may be expressed as follows: find three triangular numbers with prime indices, which form an increasing arithmetic progression.

Note that

1783, 3697, 3001 are primes congruent to 14 mod 29, that is, , and

2521, 5227, 4243 are primes congruent to 20 mod 41,

, ,

, ,

, ,

In these solutions, the numbers , , and are all primes

and, the numbers , , and form an arithmetic progression:

There are other solutions that Paul have found

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Here are a few more

757(757 + 1) – 61(61 + 1) = 1069(1069 + 1) – 757(757 + 1) = 570024

1783(1783 + 1) – 61(61 + 1) = 2521(2521 + 1) – 1783(1783 + 1) = 3177090

1657(1657 + 1) – 109(109 + 1) = 2341(2341 + 1) – 1657(1657 + 1) = 2735316

397(397 + 1) – 127(127 + 1) = 547(547 + 1) – 397(397 + 1) = 141750

1033(1033 + 1) – 331(331 + 1) = 1423(1423 + 1) – 1033(1033 + 1) = 958230

1447(1447 + 1) – 1249(1249 + 1) = 1621(1621 + 1) – 1447(1447 + 1) = 534006

2377(2377 + 1) – 1249(1249 + 1) = 3121(3121 + 1) – 2377(2377 + 1) = 4091256

2677(2677 + 1) – 1291(1291 + 1) = 3559(3559 + 1) – 2677(2677 + 1) = 5501034

Paul.

and a few more

3001(3001 + 1) – 103(103 + 1) = 4243(4243 + 1) – 3001(3001 + 1) = 8998290

3697(3697 + 1) – 127(127 + 1) = 5227(5227 + 1) – 3697(3697 + 1) = 13655250

4813(4813 + 1) – 433(433 + 1) = 6793(6793 + 1) – 4813(4813 + 1) = 22981860

4447(4447 + 1) – 727(727 + 1) = 6247(6247 + 1) – 4447(4447 + 1) = 19251000

3253(3253 + 1) – 1201(1201 + 1) = 4441(4441 + 1) – 3253(3253 + 1) = 9141660

4831(4831 + 1) – 1933(1933 + 1) = 6553(6553 + 1) – 4831(4831 + 1) = 19604970

3373(3373 + 1) – 2203(2203 + 1) = 4231(4231 + 1) – 3373(3373 + 1) = 6525090

4051(4051 + 1) – 2221(2221 + 1) = 5281(5281 + 1) – 4051(4051 + 1) = 11479590

4363(4363 + 1) – 2473(2473 + 1) = 5653(5653 + 1) – 4363(4363 + 1) = 12921930

5227(5227 + 1) – 2521(2521 + 1) = 6949(6949 + 1) – 5227(5227 + 1) = 20968794

3607(3607 + 1) – 3301(3301 + 1) = 3889(3889 + 1) – 3607(3607 + 1) = 2114154

5227(5227 + 1) – 4861(4861 + 1) = 5569(5569 + 1) – 5227(5227 + 1) = 3692574

Paul.

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