The document summarizes T. Nagell's 1960 proof of the theorem: When x is a positive integer, the number x^2 + 7 is a power of 2 only in the following five cases: x = 1, 3, 5, 11, 181. Nagell considers it necessary to publish the proof in English due to a related paper that was published. The proof proceeds by considering the Diophantine equation x^2 + 7 = 2^y in the quadratic field K(√-7) and obtaining congruences for y that lead to the five solutions for x.
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
The 1741 Goldbach [1] made his most famous contribution to mathematics with the conjecture that all even numbers can be expressed as the sum of the two primes (currently Conjecture) referred to as “all even numbers greater than 2 can be expressed as the sum-two primes” (DOI:10.13140/RG.2.2.32893.69600/1)
A Proof of Hypothesis Riemann and it is proven that apply only for equation ζ(z)=0.Also here it turns out that it does not apply in General Case.(DOI:10.13140/RG.2.2.10888.34563/5)
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
Integrability and weak diffraction in a two-particle Bose-Hubbard model jiang-min zhang
We report a bound state, which is embedded in the continuum spectrum, of the one-dimensional two-particle (Bose or Fermion) Hubbard model with an impurity potential. The state has the Bethe-ansatz form, although this model is nonintegrable. Moreover, for a wide region in parameter space, its energy is located in the continuum band. A remarkable advantage of this state with respect to similar states in other systems is the simple analytical form of the wave function and eigenvalue. This state can be tuned in and out of the continuum continuously.
The Euler-Mascheroni constant is calculated by three novel representations over these sets respectively: 1) Turán moments, 2) coefficients of Jensen polynomials for the Taylor series of the Riemann Xi function at s = 1/2 + i.t and 3) even coefficients of the Riemann Xi function around s = 1/2.
The 1741 Goldbach [1] made his most famous contribution to mathematics with the conjecture that all even numbers can be expressed as the sum of the two primes (currently Conjecture) referred to as “all even numbers greater than 2 can be expressed as the sum-two primes” (DOI:10.13140/RG.2.2.32893.69600/1)
A Proof of Hypothesis Riemann and it is proven that apply only for equation ζ(z)=0.Also here it turns out that it does not apply in General Case.(DOI:10.13140/RG.2.2.10888.34563/5)
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
Integrability and weak diffraction in a two-particle Bose-Hubbard model jiang-min zhang
We report a bound state, which is embedded in the continuum spectrum, of the one-dimensional two-particle (Bose or Fermion) Hubbard model with an impurity potential. The state has the Bethe-ansatz form, although this model is nonintegrable. Moreover, for a wide region in parameter space, its energy is located in the continuum band. A remarkable advantage of this state with respect to similar states in other systems is the simple analytical form of the wave function and eigenvalue. This state can be tuned in and out of the continuum continuously.
The Euler-Mascheroni constant is calculated by three novel representations over these sets respectively: 1) Turán moments, 2) coefficients of Jensen polynomials for the Taylor series of the Riemann Xi function at s = 1/2 + i.t and 3) even coefficients of the Riemann Xi function around s = 1/2.
Let G be a simple graph with n vertices, and λ1, · · · , λn be the eigenvalues of its adjacent matrix. The Estrada index of G is a graph invariant, defined as EE = i e n i 1 ,, is proposed as a measure of branching in alkanes. In this paper, we obtain two candidates which have the fourth largest EE among trees with n vertices
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
1. ARKIV FOR MATEMATIK Band 4 nr 13
Read 13 January 1960
The Diophantine Equation x 2+ 7 -- 2"
By T. NAGELL
In Vol. 30 of the Norsk Matematisk Tidsskrift, pp. 62-64, Oslo 1948, I published
a proof of the following theorem: 1
When x is a positive integer, the number x ~+ 7 is a power of 2 only in the/ollowing
[ive cases: x = 1, 3, 5, 11, 181.
Since prof. L.J. Mordell drew my attention to a paper by Chowla, Lewis and
Skolem in the Proceedings of the American Mathematical Society, Vol. 10 (1959),
p. 663-669, on the same subject, I consider it necessary to publish in English
my proof of 1948 which is quite elementary.
The problem consists in determining all the positive integers x and y which
satisfy the relation
1 @2+7 )=2u" (1)
4
It is evident that the difference of two integral squares u2 and v2 is equal to 7
only for u2= 16 and v2=9. Hence we conclude that the exponent y in (1) can
be even only for y=2 and x=3. Thus we may suppose that y is odd and >3.
Passing to the qtiadratic field K(]/---~), in which factorization is unique, we
get from (1)
x-+~-(!+l/=7)y ' 2 2 (2)
whence
Considering this equation modulo
2
we get, since y is odd and >=3, and since
(I+V V--5 1
1 The theorem is set as a problem in my Introduction to Number Theory, Stockholm and
New York 1951 (Problem 165, p. 2'/2).
12:2 185
2. T. NACELL,The Diophantine Equation x2+ 7=2n
the congruence
1+ ~/--7 + }/_~ (mod -3 2~/-~7)
2
But this congruence is possible only when the right-hand side is - 1/~ 7. Hence,
we must take the lower sign in (3). Thus equation (3) may be written
-1
This equation implies the congruence
-2v-l--~y (mod 7),
which has the solutions
y-----3, 5, 13 (rood 42).
Suppose first y--= 3 (mod 42) and put
y-3=TZ.6.h,
where h is an integer not divisible by 7. The number
( ) Y(Y-I)(y-2)(y-3)'(Y-4) "Tk
2[+1 "7k=(2k-2)(2k-1)2k(2k+l) 2k-3
is divisible by 7*+1 for k>2, since 7k-~>2k+l.
Hence we get from (4)
and thus
7
- 2~-~ ~ y - ~y (y- 1) (y- 2) (mod 7~+~),
--2 ~-1~-4~y-7 (modT~+l).
But this implies
y- 3 --=0 (mod 7~+1),
which is contrary to our hypothesis on y. Thus the only possibility is y = 3 cor-
responding to x = 5.
Suppose next that y ~ 5 (mod 42) and put
y-5=7~.6.h,
where h is an integer not divisible by 7. The number
2 1 (2k-4) (2k-3)... (2k+ 1)
is divisible by 7z+l for k>=3, since 7k-l>2k+ 1.
186
3. ARKIV FOR MATEMATIK. Bd 4 nr 13
Hence we get from (4)
- 2y-1 --~ - 16 ~--y - 70 + 49 (mod 7~+1).
But this implies
y - 5 ~ 0 (mod 7'+1)
which is contrary to our hypothesis on y. Thus the only possibility is y = 5 cor-
responding to x = 11.
Finally consider the case y --13 (mod 42) and put
y- 13=7 ~-6-h,
where h is an integer not divisible by 7. The number
( ) "7k- y(y-1)'''(y-13) .~y-14~.7k
2[+1 (2k- 12) ... (2k+ 1) 2k- 13]
is divisible by 7~+1 for k>7, since 7k-2>2k+ 1. Hence we get from (4)
--212------4096 --y--(Y)" 7§ (Y)'72- (Y)" 73§ (Y)'74- (Y1)" 75§
We may replace y by 13 in all the terms on the right-hand side which are di-
visible by 7. Then we get the congruence
- 4096 =- y - 4109 (mod 7~+1).
Hence
y - 13 -----0 (mod 7z+1),
which is contrary to our hypothesis on y. Thus the only possibility is y= 13,
corresponding to x= 181.
Trycktden5augusti1960
Uppsala1960.Almqvist&WiksellsBoktryckeriAB
187