On the non-linear diophantine equation 379x+397y=z2

Author(s): Sudhanshu Aggarwal1, Nidhi Sharma2
1Department of Mathematics, National P. G. College, Barhalganj, Gorakhpur-273402, U. P., India.
2Indian Institute of Technology Roorkee-247667, U. K., India.
Copyright © Sudhanshu Aggarwal, Nidhi Sharma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this article, authors discussed the existence of solution of non-linear diophantine equation 379x+397y=z2, where x,y,z are non-negative integers. Results show that the considered non-linear diophantine equation has no non-negative integer solution.

Keywords: Prime number, diophantine equation, solution, integers.

1. Introduction

Diophantine equations are those equations which are to be solved in integers. Diophantine equations are very important equations of theory of numbers and have many important applications in algebra, analytical geometry and trigonometry. These equations give us an idea to prove the existence of irrational numbers [1,2]. Acu [3] studied the diophantine equation 2x+5y=z2 and proved that {x=3,y=0,z=3 } and {x=2,y=1,z=3 } are the solutions of this equation. Kumar et al., [4] considered the non-linear diophantine equations 61x+67y=z2 and 67x+73y=z2. They showed that these equations have no non-negative integer solution. Kumar et al. [5] studied the non-linear diophantine equations 31x+41y=z2 and 61x+71y=z2 and determined that these equations have no non-negative integer solution. Rabago [6] discussed the open problem given by B. Sroysang. He showed that the diophantine equation 8x+py=z2, where x,y,z are positive integers has only three solutions namely {x=1,y=1,z=5 }, {x=2,y=1,z=9 } and {x=3,y=1,z=23 } for p=17. The diophantine equations 8x+19y=z2 and 8x+13y=z2 were studied by Sroysang [7,8]. He proved that these equations have a unique non-negative integer solution namely {x=1,y=0,z=3 }. Sroysang [9] proved that the diophantine equation 31x+32y=z2 has no non-negative integer solution.

The main aim of this article is to discuss the existence of solution of non-linear diophantine equation 379x+397y=z2, where x,y,z are non-negative integers.

2. Preliminary

Lemma 1. The non-linear diophantine equation 379x+1=z2, where x,z are non-negative integers, has no solution in non-negative integers.

Proof. Since 379 is an odd prime, so 379x is an odd number for all non-negative integer x, which implies 379x+1=z2 is an even number for all non-negative integer x, so z is an even number. Hence

z2=0(mod3)orz2=1(mod3).
(1)
Now, 379=1(mod3), which implies 379x=1(mod3), for all non-negative integer x, so 379x+1=2(mod3), for all non-negative integer x. Hence
z2=2(mod3).
(2)
Equation (2) contradicts Equation (1). Hence non-linear diophantine equation 379x+1=z2 has no non-negative integer solution.

Lemma 2. The non-linear Diophantine equation 397y+1=z2, where y,z are for all non-negative integers, has no solution in non-negative integers.

Proof. Since 397 is an odd prime, so 397y is an odd number for all non-negative integer y, which implies 397y+1=z2 is an even number for all non-negative integer y, so z is an even number. Hence

z2=0(mod3)orz2=1(mod3).
(3)
Now, 397=1(mod3), implies 397y=1(mod3), for all non-negative integer y, so 397y+1=2(mod3), for all non-negative integer y. Hence
z2=2(mod3).
(4)
Equation (4) contradicts Equation (3). Hence non-linear Diophantine equation 397y+1=z2 has no non-negative integer solution.

Theorem 1(Main theorem). The non-linear diophantine equation 379x+397y=z2, where x,y,z are non-negative integers, has no solution in non-negative integers.

Proof. There are following four cases;

  • Case 1. If x=0 then the non-linear diophantine equation 379x+397y=z2 becomes 1+397y=z2, which has no non-negative integer solution by Lemma 2.
  • Case 2. If y=0 then the non-linear diophantine equation 379x+397y=z2 becomes 379x+1=z2, which has no non-negative integer solution by Lemma 1.
  • Case 3. If x,y are positive integers, then 379x,397y are odd numbers, implies 379x+397y=z2 is an even number, so z is an even number, Hence
    z2=0(mod3)orz2=1(mod3).
    (5)
    &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp Now, 379=1(mod3), implies 379x=1(mod3) and 397y=1(mod3), so 379x+397y=2(mod3). Hence
    z2=2(mod3).
    (6)
    &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp Equation (6) contradicts Equation (5). Hence non-linear Diophantine equation 379x+397y=z2 has no non-negative integer solution.
  • Case 4. If x,y=0, then 379x+397y=1+1=2=z2, which is not possible because z is a non-negative integer. Hence diophantine equation 379x+397y=z2 has no non-negative integer solution.

3. Conclusion

In this article, authors successfully discussed the solution of non-linear diophantine equation 379x+397y=z2, where x,y,z are non-negative integers and determined that this non-linear equation has no non-negative integer solution.

Authorcontributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflict of Interests

”The authors declare no conflict of interest.”

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