Rassias hypothesis

The Rassias hypothesis is an open problem with primes . The hypothesis was expressed by Michael T. Rassias when he was preparing for the International Mathematical Olympiad ^[1] ^[2] ^[3] ^[4] ^[5] ^[6] ^[7] . The hypothesis states the following:

For any simple

p>2

{\ displaystyle p> 2}

p> 2

there are two simple

p_{1}<p_{2}

{\ displaystyle p_ {1} <p_ {2}}

{\ displaystyle p_ {1} <p_ {2}}

such that

p={\frac {p_{1}+p_{2}+1}{p_{1}}}

{\ displaystyle p = {\ frac {p_ {1} + p_ {2} +1} {p_ {1}}}}

{\ displaystyle p = {\ frac {p_ {1} + p_ {2} +1} {p_ {1}}}}

Relationship with other open issues

Rassias's hypothesis can be formulated in equivalent form:

For any prime

p>2

{\ displaystyle p> 2}

there are simple

p_{1}<p_{2}

{\ displaystyle p_ {1} <p_ {2}}

such that

p_{2}=(p-1)p_{1}-1.

{\ displaystyle p_ {2} = (p-1) p_ {1} -1.}

This reformulation shows that the hypothesis is a combination of the problem of generalized numbers Sophie Germain

p_{2}=2ap_{1}-1

{\ displaystyle p_ {2} = 2ap_ {1} -1}

with the additional condition that $2a+1$ ${\ displaystyle 2a + 1}$ should also be simple ^[3] ^[4] . This makes the hypothesis a special case of the Dickson hypothesis . Note that the Dixon conjecture (and its generalization, Conjecture H ) appeared before the Rassias conjecture. See the Preface by Pre-Michailescu ^[7] for a comparison of Rassias's hypothesis with other well-known hypotheses and open problems in number theory .

Cunningham sequences are also associated with the hypothesis, i.e. sequences of simple $p_{i+1}=mp_{i}+n,\ i=1,2,\ldots ,k-1,$ ${\ displaystyle p_ {i + 1} = mp_ {i} + n, \ i = 1,2, \ ldots, k-1,}$ for fixed mutually prime positive integers $m,n>1$ ${\ displaystyle m, n> 1}$ . Unlike the breakthrough of Ben Green and Terence Tao ^[8] on simple arithmetic progressions , the results on large Cunningham sequences are not known. Rassias's hypothesis is equivalent to the existence of Cunningham sequences with parameters $2a,-1$ ${\ displaystyle 2a, -1}$ for $a$ ${\ displaystyle a}$ such that $2a-1=p$ ${\ displaystyle 2a-1 = p}$ is simple ^[3] ^[4] .

Notes

↑ Andreescu, Andrica, 2009 , p. 12.
↑ Balzarotti, Lava, 2013 , p. 140–141.
↑ ¹ ² ³ Mihăilescu, 2011 , p. 45–47.
↑ ¹ ² ³ Mihăilescu, 2014 , p. 13-16.
↑ Rassias, 2005 , p. 885.
↑ Rassias, 2007 , p. 47.
↑ ¹ ² Rassias, 2011 .
↑ Green, Tao, 2008 , p. 481-547.

Literature

Andreescu T., Andrica D. Number Theory: Structures, Examples and Problems. - Birkhäuser, Boston, Basel, 2009 .-- S. 12.
Balzarotti G., Lava PP La Derivata Arithmetica. - Editore Ulrico Hoepli Milano, 2013. - S. 140–141.
Preda Mihăilescu. Book Review // Newsletter of the European Math. Soc .. - 2011 .-- T. 79 . - S. 45–47 .
Preda Mihăilescu. On some conjectures in Additive Number Theory // Newsletter of the European Math. Soc .. - 2014 .-- T. 92 . - S. 13–16 .
Rassias M. Th. Open Problem No. 1825 // Octogon Mathematical Magazine. - 2005 .-- T. 13 . - S. 885 .
Rassias M. Th. Problem 25 // Newsletter of the European Math. Soc .. - 2007 .-- T. 65 . - S. 47 .
Rassias M. Th. Problem-Solving and Selected Topics in Number Theory. - Springer, 2011 .-- C. xi – xiii, 82.
Ben Green, Terence Tao. The primes contain arbitrarily long arithmetic progressions // Annals of Mathematics. - 2008 .-- T. 2 . - S. 481-547 . - DOI : 10.4007 / annals.2008.167.481 .

[_9ab30cd170a00edf-1] Andreescu, Andrica, 2009 , p. 12.

[_b02f73732e6b4c49-2] Balzarotti, Lava, 2013 , p. 140–141.

[_57fa1cd1f1947e82-3] ¹ ² ³ Mihăilescu, 2011 , p. 45–47.

[_e537592812648840-4] ¹ ² ³ Mihăilescu, 2014 , p. 13-16.

[_964de5b99658c7fb-5] Rassias, 2005 , p. 885.

[_dff7fa1cc27b1bc3-6] Rassias, 2007 , p. 47.

[_4d621bea9d9fd029-7] ¹ ² Rassias, 2011 .

[_2e07498483e38fd2-8] Green, Tao, 2008 , p. 481-547.

Rassias hypothesis

Relationship with other open issues

Notes

Literature

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