Erratum to “Solutions of Indefinite Equations”, [Advances in Pure Mathematics Vol. 10, No. 9, (2020) 540-544] ()

Zengyong Liang^{}

MCHH of Guangxi, Nanning, China.

**DOI: **10.4236/apm.2022.124023
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MCHH of Guangxi, Nanning, China.

The original online version of this article (Zengyong Liang (2020) Solutions of Indefinite Equations, Volume 10(9), 540-544, doi: https://doi.org/10.4236/apm.2020.109033) unfortunately contains some mistakes. The author wishes to correct the errors. Sections 5, 6, 7, and 8 are supplemented here.

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Liang, Z. (2022) Erratum to “Solutions of Indefinite Equations”, [Advances in Pure Mathematics Vol. 10, No. 9, (2020) 540-544]. *Advances in Pure Mathematics*, **12**, 306-307. doi: 10.4236/apm.2022.124023.

5. L-Algorithm

The specific steps of the L-algorithm (three-step method) are as follows:

1) First, find out the original equation model which is lower than the original equation (or a new equation is formed by the sum value *L*(*f*) of the left term of the equation, and the unknown number of the equation is set to a smaller value), as shown in the following equation: *L*(*f*) = *w*.

For example, suppose the original equation has three terms:

${a}^{x}+{b}^{y}={c}^{u}$ , (21)

then $L\left(f\right)={a}^{x}+{b}^{y}=w$ .* *

Then:

$\left({a}^{x}+{b}^{y}\right){w}^{xy}=w{w}^{xy}$ (22)

Now, we can determine *a* = *aw ^{y}*,

6. Higher Order Indefinite Equation with Coefficients

Suppose there is a problem, find:

$k{a}^{5}+h{b}^{4}={c}^{3}$ (23)

L-algorithm is also used, for example: *k* = 3, *h* = 5. Let *a* = 3, *b* = 5, then *w* = 3854, *w ^{x}^{y}* = 3854

Generally, there is:

${k}_{1}{a}^{x}+{k}_{2}{b}^{y}+{k}_{3}{c}^{z}={k}_{4}{d}^{u}$ (24)

Obviously, *u* and *x*, *y*, *z* are mutually prime, and there is a solution using the L-algorithm using this method flexibly, more types of higher-order indefinite equations can be solved.

7. Determination of Non Solution of Indefinite Equation

Example: To prove that no odd perfect number.

Proof. The condition of even perfect number is that ${2}^{i+1}-1$ is prime. The structural equation of perfect number is derived ${2}^{i+1}-1$ = *p*, and ${2}^{i}\left({2}^{i+1}-1\right)$ is perfect number.

If there is odd prefect number, *ο*(*n*) = *sn*. Let $1+q+\xb7\xb7\xb7+{q}^{i}=p$ ,* p *is odd.

Because:

$1+q+\xb7\xb7\xb7+{q}^{i}+p+p\left(q+\xb7\xb7\xb7+{q}^{i}\right)=sp$ (25)

*s* does not contain factors of *q*, *q*^{2}, … , *q ^{i}*, then solution of (25) does not satisfy the requirement of perfect number. In addition,

1) If the *q *is not 2, $p\left(1+1+q+{q}^{2}+\xb7\xb7\xb7+{q}^{i}\right)$ can’t be factor on the left.* *

2) If the equation is not like (25), then this equation may not be established.

In any case, there is that no odd perfect number.

8. Analysis and Discussion

Birch and Swinnerton-Dyer Conjecture

Birch and Swinnerton-Dyer conjectured: “mathematicians are always fascinated by the characterization of all integer solutions of algebraic equations such as *x*^{2} + *y*^{2} = *Z*^{2}. Euclid once gave a complete solution to this equation, but for more complex equations, it becomes extremely difficult [7] .” Now, we have been able to find all integer solutions to equation of the form *a ^{x}* +

9. Conclusion

Liang, Z.Y. (2022) Erratum to “Solutions of Indefinite Equations”, [Advances in Pure Mathematics Vol. 10, No. 9, (2020) 540-544]. *Advances in Pure Mathematics*, 12, 306-307. https://doi.org/10.4236/apm.2022.124023

Conflicts of Interest

The authors declare no conflicts of interest.

[1] |
Baike (n.d.) Seven Mathematical Problems in the World. https://baike.so.com/doc/6659451-6873272.html |

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