Romanov's theorem

Romanov initially stated that he had proven the statements "In jedem Intervall (0, x) liegen mehr als ax Zahlen, welche als Summe von einer Primzahl und einer k-ten Potenz einer ganzen Zahl darstellbar sind, wo a eine gewisse positive, nur von k abhängige Konstante bedeutet" and "In jedem Intervall (0, x) liegen mehr als bx Zahlen, weiche als Summe von einer Primzahl und einer Potenz von a darstellbar sind. Hier ist a eine gegebene ganze Zahl und b eine positive Konstante, welche nur von a abhängt".{{Cite journal|last=Romanoff|first=N. P.|date=1934-12-01|title=Über einige Sätze der additiven Zahlentheorie|journal=Mathematische Annalen|language=de|volume=109|issue=1|pages=668–678|doi=10.1007/BF01449161|s2cid=119938116|issn=1432-1807}} These statements translate to "In every interval $(0,x)$ there are more than $\backslash alpha\; x$ numbers which can be represented as the sum of a prime number and a {{Mvar|k}}-th power of an integer, where $\backslash alpha$ is a certain positive constant that is only dependent on {{Mvar|k}}" and "In every interval $(0,x)$ there are more than $\backslash beta\; x$ numbers which can be represented as the sum of a prime number and a power of {{Mvar|a}}. Here {{Mvar|a}} is a given integer and $\backslash beta$ is a positive constant that only depends on {{Mvar|a}}" respectively. The second statement is generally accepted as the Romanov's theorem, for example in Nathanson's book.{{Cite book|last=Nathanson|first=Melvyn B.|url=https://books.google.com/books?id=nbjVBwAAQBAJ&dq=%22Romanov's+Theorem%22&pg=PR7|title=Additive Number Theory The Classical Bases|date=2013-03-14|publisher=Springer Science & Business Media|isbn=978-1-4757-3845-2|language=en}}

Precisely, let $d(x)=\backslash frac\{\backslash left\backslash vert\; \backslash \{n\backslash le\; x:n=p+2^k,p\backslash \; \backslash textrm\{prime,\}\backslash \; k\backslash in\backslash N\backslash \}\; \backslash right\backslash vert\}\{x\}$ and let $\backslash underline\{d\}=\backslash liminf\_\{x\backslash to\backslash infty\}d(x)$, $\backslash overline\{d\}=\backslash limsup\_\{x\backslash to\backslash infty\}d(x)$. Then Romanov's theorem asserts that $\backslash underline\{d\}>0$.{{Cite journal|last1=Elsholtz|first1=Christian|last2=Schlage-Puchta|first2=Jan-Christoph|date=2018-04-01|title=On Romanov's constant|journal=Mathematische Zeitschrift|language=en|volume=288|issue=3|pages=713–724|doi=10.1007/s00209-017-1908-x|s2cid=125994504|issn=1432-1823}}

Alphonse de Polignac wrote in 1849 that every odd number larger than 3 can be written as the sum of an odd prime and a power of 2. (He soon noticed a counterexample, namely 959.){{cite journal|last1=de Polignac|first1=A.|date=1849|title=Recherches nouvelles sur les nombres premiers|trans-title=New research on prime numbers|url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015035450967&view=1up&seq=411|journal=Comptes rendus|language=fr|volume=29|pages=397–401}} This corresponds to the case of $a=2$ in the original statement. The counterexample of 959 was, in fact, also mentioned in Euler's letter to Christian Goldbach,L. Euler, [http://eulerarchive.maa.org/correspondence/letters/OO0879.pdf Letter to Goldbach]. 16-12-1752. but they were working in the opposite direction, trying to find odd numbers that cannot be expressed in the form.

In 1934, Romanov proved the theorem. The positive constant $\backslash beta$ mentioned in the case $a=2$ was later known as Romanov's constant.{{Cite journal|last=Pintz|first=János|authorlink=János Pintz|date=2006-07-01|title=A note on Romanov's constant|journal=Acta Mathematica Hungarica|language=en|volume=112|issue=1|pages=1–14|doi=10.1007/s10474-006-0060-6|doi-access=|issn=1588-2632}} Various estimates on the constant, as well as $\backslash overline\{d\}$, has been made. The history of such refinements are listed below. In particular, since $\backslash overline\{d\}$ is shown to be less than 0.5 this implies that the odd numbers that cannot be expressed this way has positive lower asymptotic density.

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Analogous results of Romanov's theorem has been proven in number fields by Riegel in 1961.{{Cite journal|last=Rieger|first=G. J.|date=1961-02-01|title=Verallgemeinerung zweier Sätze von Romanov aus der additiven Zahlentheorie|journal=Mathematische Annalen|language=de|volume=144|issue=1|pages=49–55|doi=10.1007/BF01396540|s2cid=121911723|issn=1432-1807}} In 2015, the theorem was also proven for polynomials in finite fields.{{cite arXiv|last1=Shparlinski|first1=Igor E.|last2=Weingartner|first2=Andreas J.|date=2015-10-30|title=An explicit polynomial analogue of Romanoff's theorem|class=math.NT|eprint=1510.08991}} Also in 2015, an arithmetic progression of Gaussian integers that are not expressible as the sum of a Gaussian prime and a power of {{Mvar|1+i}} is given.{{cite arXiv|last1=Madritsch|first1=Manfred G.|last2=Planitzer|first2=Stefan|date=2018-01-08|title=Romanov's Theorem in Number Fields|class=math.NT|eprint=1512.04869}}

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Category:Theorems in number theory

Category:Additive number theory