Lev Genrikhovich Schnirel'mann(Lev Genrihoviq Xnirel~man)by
V. Tikhomirov and V. Uspenskii
Kvant, 1996, No. 2. pp. 2-6
Original Russian version at
http://kvant.mirror1.mccme.ru/1996/02/lev_genrihovich_shnirelman.htm
January 1995 marks the 90>2 anniversary of the birthday of L. G. Schnirel'mann, oneof the most remarkable mathematicians of 1930's.
Until the present day we do not fully comprehend one of the most amazing miraclesof the twentieth century: The Moscow school of mathematics. In 1914 there was only one
internationally recognized mathematician working in Moscow. This was Dmitrii
Fedorovich Egorov, who was working in a narrow specialty of differential geometry. At
the same time France had giants such as Poincare, Picard, Hadamard, Lebesgue, Borel;
Germany was the home to Klein, Hilbert, Weyl, all of them mathematicians of thebroadest range. They belonged to the most distinguished schools of mathematics incenturies!
And then, so quickly, twenty years passed, seven of which were filled with wars thatdevastated our country, a blink of an eye in history. But, when in the mid-thirties, a wellknown American mathematician S. Lefshetz was asked who, in his opinion, were themost remarkable young mathematicians, he named four: Gel'fond, the solver of one of theHilbert's problems, Kolmogorov, who made outstanding contributions to the theory oftrigonometric series, probability, topology, and many other areas of mathematics,
Pontriagin, discoverer of the new areas in topology and analysis, and Schnirel'mann, who
obtained striking results in number theory, topology, and variational calculus.
In addition, one should mention the brilliant figures such as Urysohn, who died at theˆ Khinchin,
age of 26, Alexandrov, Bari, P. Novikov, Lavrentiev, Lyusternik, Petrovskii,
all of whose golden period of creativity happened during these years. Furthermore, thosewere the years when Gel'fand was taking his first steps . . .
However, in this bouquet of remarkable talent, the accomplishments of Schnirel'mann
shine the brightest.
L. G. Schnirel'mann was born on January 2, 1905, in the town of Gomel. His father
was a teacher of Russian language.
Lev Genrikhovich, early on, has shown to have an outstanding talent. He drew, wrote
poetry, and at the age of 12 independently and on his own completed a course inelementary mathematics. For a period of several months the youngster attended courses
in physics and mathematics given for high school graduates. There, he attracted attention1

of the instructors, to the point that they arranged for the youngster to be sent to Moscow
for further studies.

Lev Genrikhovich Schnirel'mann
At the age of 15 he already flexed his muscles in an independent work. According to
the legend (these always accompany biographies of outstanding people), he arrived in2

Moscow at the age of 16 and enrolled in the Moscow (State, tr.) University. He broughtwith him notes, written in a school exercise book (made of cheap paper, better quality
was not available during those difficult times), which contained a theorem about
partitioning of a sphere. We will discuss this theorem later on, suffice it to say now is thatit played a crucial role in the solution of the Poincare problem about closed
geodesics  the solution that was found later by Schnirelman, jointly with Lazarev
Aronovitch Liusternik. Solving this problem made Schnirel'mann known throughout theworld.
Upon completing his studies at the university in two and a half years, Schnirel'mann
was accepted as an aspirant at the Institute of Mathematics and Mechanics at the MGU.
Like almost all of the mathematicians mentioned by us, he was a a student of Nikolai
Nikolaevitch Lusin, the only exceptions being Pontriagin, who was a student of
Alexandrov, and Petrovski, who was a student of Egorov.
Lazar Aronovich (Liusternik, tr.) used to reminisce that Luzin, (who apparently was
inclined to view the world in a somewhat mystical way), had a dream in which a youngman (“with the same biography as Lev Genrikhovich,” the way L. A. put it), came to him
and solved the continuum hypothesis. Then, when young Schnirel'mann actuallyappeared on the scene, Liusternik viewed him as a messenger from God. But
Schnirel'mann did not solve the continuum hypothesis, the solution of that problem had
to wait more than sixty years, when it was found by Paul Cohen. (It was actually only 35,
tr.)
Schnirel'mann published three of his most remarkable papers during two years: in1929 and 1930. Here are the statements of these theorems:
Theorem 1. Inside an arbitrary closed curve in the plane, one can inscribe a square.
Consider a thin piece of thread, with both ends tied together. This results in a “closed
curve.” We toss this curve on a table, and we are guaranteed that one can find four points
on the curve, forming the vertices of a square.
Theorem 2. On an arbitrary smooth surface of spherical type, there exists threeclosed geodesics.
Imagine in your mind that you are at the see shore and you pick up a very smoothstone. Take a very thin and very elastic rubber band and stretch it over the stone in such a
way that it “does not slip.” If you succeed, you found yourself a geodesic. On thespherical ball, the geodesics are the great circles. If the rubber band moves a bit from agreat circle it will slip off the ball. On the ellipsoid there are only three geodesics: These
are formed by intersections of the ellipsoid with the planes perpendicular to it's axes.
Poincare conjectured that “on an arbitrary smooth stone, there are at least three
geodesics.” In 1929 this conjecture was proved by Liusternik and Schnirel'mann. That
created a world wide sensation.

3

Theorem 3. There exists a natural number R , such that any natural number is a sumof at most R prime numbers.
Problem. Any natural number, larger or equal than six, can be represented as a sumof three prime numbers.
This question was presented to Euler by Christian Goldbach. He was a German
mathematician who lived half of his life in Russia and who actually died in Moscow. He
posed this question in a letter from June 7, 1742. In reply, dated June 30, 1942, Euler
showed that in order to solve this problem it is enough to show that an arbitrary evennumber, larger or equal to 4, is a sum of two primes.
The first progress in search of the solution of this problem (not solved until thepresent day) was made by Schnirel'mann. (Prior to that time, Hardy and Littlewood
published a paper in which they prove the Goldbach hypothesis (for sufficiently large
natural numbers), under an assumption of the truth of some (heretofore unproven)hypothesis. In 1937 I. M. Vinogradov proved the Goldbach conjecture for sufficiently
large natural numbers.) However, the most important thing here was not the fact that any
natural number can be represented by a sum of bounded number of prime terms. (In fact,
in the Schnirel'mann's method, the number of summands is estimated to be several
hundred thousands.) What was the most important was actually the highly original andingenious method, which can be extended to solve several other problems. We will say a
few words about this later on.
In 1931 Schnirel'mann was send abroad for three months, and the trip was atremendous success. For a period of time he was working in Göttingen  the Mecca ofmathematics in those days  where the great Hilbert lived and worked. (Schnirel'mann isremembered not only by his phenomenal results, but also by the fact that “walked
barefoot through the streets of Göttingen”  as Constance Reid wrote in her book about
Courant.) He was asked by the most prestigious German publisher to write a monograph,
but this was not to be: German government was taken over by fascists.
In 1933 Schnirel'mann was chosen to be a corresponding member of Academy of
Sciences of the USSR.
In 1934 the governing board of the Moscow Mathematical Society decided to conduct
the first high school mathematical Olympiad and L. G. Schnirel'mann was appointed tothe organizing committee. He was one of the initiators (together with Liusternik andGel'fand) of the mathematical circles at the MGU. The professors and the lecturers fromthat institution would twice a month give talks to the high school students. Schnirel'mann
was also one of the organizers of these talks. In particular, he would give lectures on thehigher dimensional geometry and group theory.
Schnirel'mann was one of the first people in Moscow to study convex geometry. He
wrote a remarkable paper on the subject of applications of convex geometry to the theoryof the best approximations. It was published posthumously.
4One should also mention one other paper of his, written jointly with L. S. Pontriagin,
on the “metric definition of dimension.” This paper had an influence on the developmentof the concept of %-entropy, carried out by Kolmogorov.
Lev Genrikhovich was very friendly with Liusternik, Gel'fond, and Gel'fand. Many
people recall him as an individual of extremely high caliber, a person who was sensitiveand delicate, who had varied intellectual interests, a person of sharp mind, a keenobserver, and who was also very spiritual.
His life ended tragically: On the 24th of September 1938 he committed suicide. The
people of the older generation, with whom we had an opportunity to discuss the matter,connect the tragedy to the bloody and senseless atmosphere of those times: They say thatNKVD became interested in Lev Genrikhovich. This made him extremely frightened,
and he decided to end his life. Perhaps the veil of mystery over those events will be lifted
one day, when someone seeking the truth will have an opportunity to look at the archivesof KGB.
We now come to a discussion of various theorems proved by Schnirel'mann. We
begin with a result from his young age, which we already mentioned.
Theorem about partition of a sphere. Suppose a sphere is painted with three
distinct colors. Thenß there exists a pair of antipodal points (i. e., diametrically opposed
points) which are painted the same color.
This formulation needs to be made a bit more precise. Painting the sphere W withthree colors means decomposing the sphere into three parts J" ß J# ß J$ ß the union ofwhich is W . Moreover, we do not require that these sets are disjoined, each point can be
simultaneously painted with several colors. However, without additional assumptions on
the sets J3 , the assertion is obviously false: One can decompose the sphere into two nonintersecting parts J" and J# such that for any pair of antipodal points Bß C, one of thepoints Bß C belongs to J" and the other to J# . However, for an arbitrary such a
decomposition, the sets J" and J# turn out to be not closed: One of the sets contains a
non-empty part of the boundary of the other. (A set J in the plane or in the space iscalled closed if it contains its boundary. Equivalently, every point not belonging to theset is located a positive distance from the set, and is not immediately adjacent to it.)Now we can give a correct formulation of the theorem about decomposition of thesphere.
Theorem 4. Suppose a sphere is covered by three closed sets. Then one of these setscontains a pair of antipodal points.
What is meant here by a sphere is the ordinary two dimensional sphere W # , located in
the three dimensional space ‘$ , and which is given by the equation B#"  B##  B#$ œ "ÞAnalogously, one defines a sphere W 8 for an arbitrary natural number 8: It is lies in the5Ð8  "Ñ  dimensional space ‘Ð8"Ñ and consists of the set of solutions of the equation" B#3 œ "8"3œ"

If ÐB" ß B# ß á ß B8 Ñ is a point of the 8-dimensional sphere, then its antipode is the pointÐ  B" ß  B# ß á ß  B8 Ñ. Schnirel'mann proved his theorem for spheres of arbitrary
dimension: If an 8-dimensional sphere is painted in 8  " colors (i. e., it is covered byclosed sets J" ß á ß J8" ), then one can find a pair of antipodal points which are painted
the same color.
Schnirel'mann's theorem is equivalent to another theorem, proved in 1930's by two
Polish mathematicians K. Borsuk and S. Ulam: Every continuous transformation 0 of thesphere W 8 into the space ‘8 ß glues together two antipodal points. Another words, there isa point B − W 8 such that 0 ÐBÑ œ 0 Ð  BÑÞ (Here, and in what follows, all thetransformations are assumed to be continuous.) One more, equivalent form of BorsukUlam theorem is this: There exists no odd transformation 0 À W 8 È W 8" . Atransformation 0 is called odd if 0 Ð  BÑ œ  0 ÐBÑ for all B'sÞ
Let us now explain why these three theorems are equivalent. We will denote them by
X (Schnirel'mann's theorem about painting of spheres), BU" (the theorem about gluingof antipodal points), and BU# (the theorem about the nonexistence of an odd
transformation). Since W 8" © ‘8 , any transformation into W 8" can be viewed as atransformation into ‘8 . In this way, the implication BU" Ê BU# is obvious.
Conversely, suppose BU# holds. Let us assume that 0 À W 8 È ‘8 is a counterexample
for BU" , i. e., 0 a transformation which does not glue antipodal points. Then, thetransformation 1 À W 8 Ä ‘8 given by the formula 1ÐBÑ œ 0 ÐBÑ  0 Ð  BÑ is odd and does
not assume the value zero. There exists a natural transformation < À ‘8 Ö!× È W 8" ,which assigns to every non-zero vector B œ ÐB" ß á ß B8 Ñ − ‘8 a unit vector <ÐBÑ which
has the same direction as B:
B<ÐBÑ œ
lBl
where lBl œ Œ!B#3  . The composition <1 À W 8 È W 8" is odd, contrary to BU# .
"#We now establish the equivalence of the Borsuk-Ulam and Schnirel'mann's theorems.
We start with BU1 and let J" ß á ß J8" be closed subsets of the sphere W 8 , such that
their union is W 8 . We must show that for some 3ß " Ÿ 3 Ÿ 8  ", the set J3 contains apair of antipodal points. If there exists a point B, belonging to all the sets J3 , the situation
is clear: Some set J3 contains the antipodal pair Bß  B. Assume now thatJ"  á  J8" is empty. Then, for each B − W 8 , let 03 ÐBÑ be the distance of the point B
to the set J3 . Hence 03 À W 8 È ‘ is a positive, non-negative function, and 03 ÐBÑ œ ! ifand only if B − J3 . (Here we use the hypothesis that the sets J3 Ñ are closed. As aconsequence of our hypothesis, the functions 03 ß " Ÿ 3 Ÿ 8  ", never assume the value6

! simultaneously, thus the function
2 œ "038"3œ"is everywhere positive. Define 13 œ 023 ß " Ÿ 3 Ÿ 8  ", andKÐBÑ œ a1" ÐBÑß á ß 18 ÐBÑbThen K À W È ‘8 is a continuous transformation. We apply the Borsuk-Ulam theorem to
K and conclude that there is B − W 8" , such that 13 ÐBÑ œ 13 Ð  BÑ for each 3 œ "ß á ß 8.
Since
18" œ "  "138"3œ"

we also have 18" ÐBÑ œ 18" Ð  BÑÞ If 3 is such that B − J3 , then 13 ÐBÑ œ 13 Ð  BÑ œ !,thus J3 contains a pair of antipodal points.
We preface the proof of the implication X Ê BU# with the following remark: In the
Schnirel'mann's theorem, the number 8  " of the colors cannot be replaced by 8  #.
Another words, the 8-dimensional sphere can be covered with a collection of closed setsJ" ß á ß J8# ß none of which contains a pair of antipodal points. Let, for example
J3 œ eaB" ß á ß B8" b − W 8 À B3

%fß 3 œ "ß á ß 8  "andJ8# œ aB" ß á ß B8" b − W À "B3 Ÿ  %Ÿ8"3œ"For sufficiently small %, the sets J" ß á J8# cover the sphere, and none of them contains
a pair of antipodal points. We are now ready to prove implication X Ê BU# . Supposethat 0 À W 8 È W 8" is an odd transformation. If follows from the previous remarks thatthe sphere W 8" can be covered by closed sets J" á ß J8" , none of which contains a pairof antipodal points. Denote by 0 " ÐEÑ to be pre-image of the set E, i. e., the set ofpoints of those B, for which 0 (BÑ − EÞ Then, 0 " ÐJ" Ñß á ß 0 " ÐJ8" Ñ is a cover of thesphere W 8 by closed sets, none of which contains a pair of antipodal points. This is
impossible by the Schnirel'mann's theorem.
Let us sketch the proof of the Borsuk-Ulam theorem (in the form BU# ) in the case
8 œ #. We will show that there does not exist an odd transformation 0 À W # È W " . By thearguments above, this will also prove the Schnirel'mann's theorem about painting a twodimensional sphere with three colors.
We remark that for 8 œ " the Borsuk-Ulam theorem (in the form BU# ) is trivial,
since the zero-dimensional sphere consists of two pints „", and a continuous

7

transformation of W " into W ! must be a constant, we cannot “tear” the circle into twoparts. We will reduce the proof of BU# for 8 œ 2 to the case 8 œ ".
Suppose a point is moving on a circle, and after a while returns to the original
position. Intuitively, it is clear what is meant by a number of complete winds(circumnavigations), or the winding number, the point makes during the entire course of
the motion (even if the movement of the point is not always in the same direction).
Formally, we can define this number as follows. Suppose the motion of the point on thecircle is described by a continuous function 0 À M È W " , where M œ Ò+ß ,Ó is an intervalon the real line. Then there exists a continuous function : À M È ‘ such that0 Ð>Ñ œ acos:Ð>Ñß sin:Ð>Ñb

for all > − M . Such a function is determined uniquely, up an addition of a constant of theform #15 . This means that the difference :Ð,Ñ  :Ð+Ñ is uniquely determined. If
0 Ð,Ñ œ 0 Ð+Ñ, then a:Ð,Ñ  :Ð+ÑbÎ#1 is an integer, which is then called the winding
number.
Suppose now we are given a continuous transformation 0 À W " È W " of a circle into
itself. We change this transformation into a transformation 1 À Ò!ß #1Ó È W " , by putting
1Ð>Ñ œ 0 Ðcos>ß sin>ÑÞ The number of winds the point 1Ð>Ñ makes when > varies from 0 to
#1 is called the degree of the transformation 0 . For example, the identity transformation
has degree ", the constant transformation has degree !, and the symmetry transformation
about any diameter of the circle has degree  ". Since the degree is an integer, it cannotchange under a continuous deformation of the transformation. (We will not prove thisassertion, nor will we give an exact definition what is a deformation of a transformation.)
Suppose H is the closed disk bounded by the circle W " Þ We have the following
Proposition 1. If the transformation 0 À W " È W " can be extended to a continuoustransformation J À H È W " , then the degree of 0 is zero.
Proof. For < − Ò!ß "Ó and B − W " set 0< ÐBÑ œ J Ð<BÑ. The transformation 0< (BÑdepends on < continuously, so that all the transformations 0< have the same degree. Since
0! is a constant transformation, it has degree zero. Consequently, the degree of 0" œ 0 is
also !.
Proposition 2. If 0 À W " È W " has degree zero, then 0 glues together some pair ofantipodal points (i. e., there is a point B − W " such that 0 ÐBÑ œ 0 Ð  BÑ).
Proof. From the hypothesis one can easily deduce that there exists a continuousfunction : À M È ‘ such that 0 ÐBÑ œ acos:ÐBÑß sin:ÐBÑb for all B − W " Þ Since : glues
together a pair of antipodal points (case 8 œ " of the Borsuk-Ulam theorem), the same istrue for 0 .
The case 8 œ # of the Ulam-Borsuk theorem follows now directly from the
Propositions 1 and 2. It is enough to show that any transformation J À W # È W " glues apair of antipodes. Consider the restriction 0 of the transformation J to the circle W " ß

8resulting from the intersection of the sphere with the horizontal plane. We identify thiscircle with W " ß and identify the resulting disk with the disk H from the Proposition 1.
Suppose : is the projection of the upper hemisphere onto H, and ; be the inverse of :.
Then, the transformation B È J Ð;ÐBÑÑ from H into W " is an extension of 0 . It follows
from the Proposition 1 that 0 has degree zero. It follows then from the Proposition 2 that0 glues a pair of antipodal points. Thus we have proved the Borsuk-Ulam-Schnirel'mann
theorem for a two-dimensional sphere.
We will now present another remarkable example, due to Schnirel'mann, of an
application of topological methods. What we have in mind is the theorem about inscribed
square. We already have encountered the statement of this theorem: In any closed curve
one can inscribe a square. We now present the idea of the proof.
Consider the space of all quadrilaterals in the plane. Since each such quadrilateral canbe described by eight numbers, we can say that this set is eight-dimensional. We now
consider two “four-dimensional” subsets E and F as follows: Each element of E and F
will be described by four parameters. The set E will consist of all squares, and the set F
will consist of all the quadrilaterals whose vertices lie on the curve. The theorem assertsthat these two sets intersect. We will begin with an assumption that the curve is anellipse. In this case, it is easy to show that there exists exactly one inscribed square. Thus,
the sets E and F , when defined for an ellipse, intersect. In the general case, one cancontinuously deform the give curve into an ellipse.

The set F will also deform continuously during that operation. One can show that in thecase of ellipse, the sets E and F intersect in a “robust”: that intersection cannot disappear
under a continuous deformation. One can make an analogy with the intersection of themeridian and the parallel on the torus (the surface of a “doughnut”  see the illustration).
Consequently, Eand F intersect for an arbitrary curve.

9Carrying out the details of this idea presents several difficulties. I order to apply the
results of the corresponding theory it is necessary to assume that the sets E and F include“degenerate” squares, i. e., those for which all the vertices collapse to a single point. How
does one avoid the possibility that the intersection E  F will contain only suchdegenerate points? For this purpose Schnirel'mann assumes that the curve is sufficiently
smooth (is actually twice differentiable). Quite often, when this Schnirelman theorem iscited, it is formulated for the general curve, without this extra hypothesis. The authors
don't know if the proof in general case was ever published.
Finally, we will describe Schnirel'mann's method in the additive number theory.
Suppose E and F are two sets of natural numbers. The sum of E and F is the set,
denoted by E  F , of all the numbers of the form +  , , where + − E, , − F . For our
purpose it will be more convenient to define the sum of the sets E and F as the set given
by E Š F œ (A+B)  E  F , i. e., the set obtained from E  F by adding all theelements of E and F . We say that a set E is a basis of the natural numbers if for some 5
the 5  fold sumE Š á Š E, coincides with the set of all natural numbers. For example,
if E is the set of all squares, then E is a basis. This follows from the well known theorem,
due to Lagrange, which says that any natural number is a sum of at most four squares.
This means that E Š E Š E Š E œ . Let T be the set of all prime numbers. Is T abasis? Schnirel'mann was the first one to give a positive answer to this question. He
proved that indeed T is a basis. We now present the main ideas of that proof.
We follow Schnirel'mann and introduce a concept of the density of a set E of natural
numbers. For each 8 − ß let EÐ8Ñ be the number of elements of the set E which are in
the interval Ò"ß 8Ó. We define the density .ÐEÑ of the set E to be the lower bound of thenumbers EÐ8ÑÎ8 taken over all 8 −  (i. e., the largest number + such that EÐ8ÑÎ8  +for an arbitrary 8). Another words, the density is the largest number !, such thatEÐ8Ñ !8 for all 8 − . Schnirel'mann proves the following result:
Theorem 5. Any set of natural numbers having a positive density is a basis.
This theorem cannot be directly applied to the set T of prime numbers, augmented by
the number ", because the resulting set has density equal to zero. (Chebyshev proved thatthe number 1Ð8Ñ, of prime numbers not exceeding 8, is less than G8Îlog8ß for some G ;See the paper by V. Tikhomorov, “On the Chebyshev's theorem about the distribution ofprime numbers”, Kvant, No. 6, 1994)
However, Schnirel'mann has proved that T Š T has a positive density, from which it
follows that T is a basis. We recall, that Euler's question whether or not the set T Š T
contains all even numbers remains open.
We will prove Theorem 5. The assertion will follow from the following two lemmas.
Lemma 1. If Eß F −  and .ÐEÑ  .ÐFÑ  ", then E Š F œ .

10

Proof. Fix 8 − . If 8 − F then 8 − E Š F . If 8 Â F then consider two subsets of
the interval Ò"ß 8Ó: Ö+ − E À + Ÿ 8× and Ö8  , À , − Fß , Ÿ 8× They are guaranteed tointersect, since the first set has at least 8 † .ÐFÑ elements, the second has at least 8 † .ÐFÑelements and 8 † .ÐEÑ  8 † .ÐFÑ  8. Consequently, + œ 8  , for some + − E,
, − F , so that 8 œ +  , − E Š F .
Lemma 2. (Schnirel'mann inequality) For arbitrary E and F §  we have thefollowing.ÐE Š FÑ.ÐEÑ  .ÐFÑ  .ÐEÑ † .ÐFÑÞ

Proof. Set G œ E Š Fß ! œ .ÐFÑ, and " œ .ÐFÑ. Fix 8 − . We have to estimate
the number GÐ8Ñ from below. Let +"  á  +< , < œ EÐ8Ñ be all the elements of the set
E which are also in the interval Ò"ß 8Ó. The interval Ò"ß 8Ó is partitioned by the numbers+" ß á ß +< into <  "subintervals (some of these may be empty) of lengths 6" œ +"  "ß6# œ +#  +" , âß 6<" œ 8  +< . The 5 >2 such subinterval contains at least " † 65 numbers
from G . For 5  " these are numbers of the form +5"  ,, where , − F , , Ÿ 65 ß and for
5 œ " these are numbers from F , that are larger than 6" . From this we deduce thefollowing estimate:GÐ8Ñ<  " † " 65 œ <  " Ð8  <Ñ œ Ð"  " Ñ<  " 8which implies that .ÐGÑÐ"  " Ñ!8  " 8

Ð"  " Ñ!  " œ !  "  !" ÞWe now prove the Theorem 5 from Lemmas 1 and 2. The inequality in Lemma 2 canbe written as"  .ÐE Š FÑ Ÿ Ð"  .ÐEÑÑ ‚ Ð"  .ÐFÑÑ
In this form it can be extended (by induction) to the arbitrary number of terms:"  .ÐE" Š á Š E8 Ñ Ÿ $Ð"  .ÐE3 ÑÑÞ
83œ"Suppose now E is a set with positive density andE5 œ E Š á Š E
be the sum of 5 terms, each one equal to E. The last inequality shows that .ÐE5 Ñ
approaches 1 with increasing 5 . Let 5 be such that .ÐEÑ  "Î#. It follows from Lemma 1
that E#5 œ Þ This proves Theorem 5.
For every 8 −  let [ œ Ö"8 ß #8 ß á × be the set of all the 8>2 powers. Is [8 a basis?
This is what's called the Waring's problem. The question was answered in a positive way
by Hilbert at the beginning of (the 20>2 , tr.) century. The solution turned out to be quite
complicated. Theorem 5 allows one to obtain another solution: It is sufficient to showthat the 5 -fold sum [8 Š á Š [8 is of positive density for sufficiently large 5 . An11elementary (but very complicated) solution of the Waring's problem, based on the
Schnirel'mann method, can be found in the book by Khinchine [1].
In conclusion, we make several remarks.
1. From Proposition 1 we can deduce the following result:
Brouwer's Theorem. There does not exist a continuous map J of the disk H into thecircle W " which fixes the points on the boundary, such that J ÐBÑ œ B for all B − W " .
Indeed, the identity transformation on the circle bounding the disk has degree ",hence it cannot be extended to the mapping J À H È W " . From this one can easily
deduce the theorem about a fixed point: for any transformation J À H È H there is
B − H such that J ÐBÑ œ BÞ
The concept of degree of the transformation can be extended to a transformation of
the 8-dimensional sphere, and one can use this concept to prove the fixed point theorem
for the Ð8  "Ñ-dimensional ball. This was done by Brouwer in the early part of thiscentury (20>2 , tr.) and the result was a spectacular achievement of the emerging field ofmathematics  topology.
2. A well known geometer Boris Nikolaevich Delone, when commenting on the
Schnirelman's theorem about the squares inscribed in a curve, noticed that if the curve inquestion is convex, the theorem can be proved by elementary methods. We encourage
you to try to find such an elementary proof for yourself.
3. Suppose E œ Ö"ß %ß *ß "'ß á × be the sequence of all the squares of the natural
numbers. We remarked that E Š E Š E Š E œ . Do you know what are the sets E Š E
and E Š E Š E? The first set actually consists of all the integers, whose factorization
into primes has the property that all the primes of the form %5  $ occurring in thefactorization, appear with an even exponent. The second set is obtained from the first byadding all the numbers of the form %+ Ð),  (ÑÞ4. In connection with Lemma 2 let us quote (with some abbreviations) from the book[1]: “In the fall of 1931 when Schnirel'mann talked about his discussion in Göttingenwith E. Landau, he reported that he established the following interesting fact: For all theexamples he could think of, the inequality.ÐE Š FÑ.ÐEÑ  .ÐFÑ  .ÐEÑ.ÐFÑ

can be replaced with a stronger, and simpler inequality:.ÐE Š FÑ.ÐEÑ  .ÐFÑ(provided that .ÐEÑ  .ÐFÑ Ÿ "Ñ. The first attempts to prove this conjecture were
unsuccessful. The problem became fashionable. The mathematical community was
fascinated by it. A good half of the English mathematicians put aside whatever they were12

doing and were working on trying to find the proof. But, the problem turned out to bequite stubborn, and for many it years resisted all the attempts, some of which used themost sophisticated of methods. Only in 1942 it was finally cracked by a young Americanmathematician Mann.”
The proof of the Landau-Schnirel'mann conjecture can be found in the Khinchin'sbook [1]. We strongly encourage the readers to get acquainted with this remarkable book.
Equally deserving your attention is the book by Schnirel'mann himself [2]. You canfind there the proof of Lagrange's theorem about the sum of four squares, the proofs ofthe great Fermat theorem for the exponents 3 and 4, as well as many other things.
References
1. Khinchin, A. Ja., Three pearls of number theory
2. Schnirel'mann, L. G., Prime numbers.

Translated by Vladimir Drobot
Retired and gainfully unemployed
http://www.vdrobot.com
mailto:drobot@pacbell.net

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