# Patterns of Primes in Arithmetic Progressions

## Abstract

After the proof of Zhang about the existence of infinitely many bounded gaps between consecutive primes the author showed the existence of a bounded *d* such that there are arbitrarily long arithmetic progressions of primes with the property that *p*^{ ′ } = *p* + *d* is the prime following *p* for each element of the progression. This was a common generalization of the results of Zhang and Green-Tao. In the present work it is shown that for every *m* we have a bounded *m*-tuple of primes such that this configuration (i.e. the integer translates of this *m*-tuple) appear as arbitrarily long arithmetic progressions in the sequence of all primes. In fact we show that this is true for a positive proportion of all *m*-tuples. This is a common generalization of the celebrated works of Green-Tao and Maynard/Tao.

## Notes

### Acknowledgements

The author thanks to the referee for his valuable suggestions. The author was supported by National Research, Development and Innovation Office, NKFIH, OTKA NK 104181, K100291 and ERC-AdG. 321104.

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