Introduction

Although the Legendre’s conjecture is less famous than some other conjectures such as the Goldbach’s conjecture, the Fermat’s last theorem, the Riemann’s conjecture and so on, it is also one of the puzzling problems which has not been solved by mathematicians all over the world. The Legendre’s conjecture was proposed by Adrien-Marie Legendre in the time between 1752 and 1833, but no one has given a satisfactory proof of it. From the long time ago to the recent years, the Legendre’s conjecture has become one of the interesting topics in the fields of mathematics^1-5. Although people published a lot of papers about the Legendre’s conjecture, their theories are complex and not easy for people to understand. The previous proofs on the Legendre’s conjecture have not been satisfactorily accepted by the mathematicians over the world. Using the very simple mathematical theory, this paper will present a concise proof of the Legendre’s conjecture.

Materials and Method

The Legendre’s conjecture can be described as the following theorem.

Theorem: With respect to an arbitrary integer n > 1, there must be a prime P such that n² < p < (n + 1)².

The proof: At first, if the integer n² is not very big, it is easy to test them one by one, for example,

When n = 1 (n + 1)² = 4 there are primes 2 and 3 such that

1 < 2,3 < 4. Successively, there are primes 5 and 7 such that 2² < 5,7 < 3²; similarly, 3² < 11,13 < 7²; 4² < 17,19,23 < 5²; 5² < 29,31 < 6²; 6² < 37,41,43,47 < 7²; 7² < 53,59,61 < 8²; 8² < 67,71,73,79 < 9²; 9² < 5,7 < 10²; 10² < 101,103,107,109,113 < 11²; 11² < 127,131,137,139 < 12²; 12² < 149,151,157,163,167 <13²; 13² < 173,179,181,191,193 < 14²; 14² < 197,199,211,223 < 15²; 15² < 227,229,233,239,241,251 < 16²; when n = 16, 16² < 257,263,269,271,277,281,283 <17²; 17² < 293,307,311,313,317 < 18²; 18² < 331,337,347,349,353,359 < 19²; 19² < 367,373,379,383,389,397 < 20²; when n = 20, 20² < 401,409,419,421,431,433,439 < 21²; 21² < 443,449,457,461,463,467,479 < 22²;  22² < 487,491,499,503,509,521,523 < 23²; when n=23, 23² < 541,547,557,563,569,571 < 24² ; when n=24, 24² < 577,587,593,599,601,607,613,617,619 < 25²……. Up till to n = 315, n² = 99225, (n + 1)² = 99856, there are primes 99233, 99241, 99251, 99257,……99829, 99833, 99839 such that 99225 <  99233, 99241, 99251, 99257,……99829 , 99833, 99839< 99856. The Legendre’s conjecture is true for all of the above examples.

Anyway, if the big number n² > 99856, it is reasonable to use the prime number theorem for the proof, because this theorem is very correct for big positive integers. According to the prime number theorem ^6-7, when n is a big number, the number of primes smaller than n is .

Results

According to the prime number theorem, the number of primes smaller than (n+1)² is given by N₁ = (n+1)² / ln(n+1)² , the number of primes smaller than n² is given by N₂ = n² / lnn² When n² is very big, if

there must be a prime p such that n² < p < (n+1)². The case of all the integers bigger than n² can be approximately regarded as n² - oo, it is equivalent of that n - oo.

In consideration of

is very big, or if n = oo,

This is a limit of  oo/oo type, using the L. Hospital’s law of solving the limit, respectively from the derivatives of the numerator and denominator of the fraction in eq. (1), eq. (1) becomes

From eq. (1) and eq. (2) it demonstrates that when n² is very big or

it has proven the Legendre’s conjecture.

Discussion

With respect to the Legendre’s conjecture, the paper proved that it is true for a lot of finite positive integers. Moreover, for infinite very big positive integers which, for example, are bigger than n² > 99856, they can be approximately regarded as n² -oo, or n - oo, the limit is a oo/oo using the prime number theorem and the L . Hospital’s law , from eq .(1),

it gets that

this demonstrates that there must be at least one prime p , such that n² < p < (n + 1)², n is an arbitrary big positive integer , therefore, the paper has proven the Legendre’s conjecture.

Conclusion

The paper proved the Legendre’s conjecture by use of the prime number theorem. In the proof, because the prime number theorem holds true for big numbers, therefore, the paper tested the conjecture with small integers one by one from n = 2, as shown as the paper presented in the front. But we cannot infinitely test all the integers, when the integer n² is big, according to the prime theorem, the number of primes smaller than (n + 1)² is equal to N₁ (n+1)²/ ln (n + 1)²; and the number of primes smaller than n² is equal to N₂ = n² / lnn². It is easy to understand that if N₁ - N₂ > 0, or N₁/N₂ >1, there must be one or more primes between N₂ and N₁, therefore, the Legendre’s conjecture is true. The paper concisely proved this conjecture.

Acknowledgement

The author thanks Hunan Normal university (present address) provided an opportunity of research work to me, to let me be interesting in studying this subject.

Funding Sources

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Conflict of Interest

The authors do not have any conflict of interest.

Data Availability Statement

This statement does not apply to this article

Ethics Statement

This research did not involve human participants, animal subjects, or any material that requires ethical approval.

Informed Consent Statement

This study did not involve human participants, and therefore, informed consent was not required.

Permission to reproduce material from other Sources

My paper is completely new, no reproduce material from other sources.

Author Contributions

The author finished all the works by himself.

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