April 2004 JNT Paper On $p^x - q^y = c$ and related three term exponential Diophantine equations with prime bases

2006 JNT Paper On the generalized Pillai equation $\pm a^x \pm b^y = c$.

June 2011 JNT Paper The generalized Pillai equation $\pm r a^x \pm s b^y = c$

2013 JTNBordeaux Paper(Theoretical) The number of solutions to the generalized Pillai equation $\pm r a^x \pm s b^y = c$

*Integers*: The Electronic Journal of Combinatorial Number Theory,
2014 (Computational)
Computations for "Handling a large bound for a problem on the generalized Pillai equation $\pm r
a^x \pm s b^y = c$"

Bennett's Pillai theorem with fractional bases and
negative exponents allowed :
Reese Scott; Robert
Styer

Bennett’s Pillai theorem with fractional bases and negative exponents allowed

Journal de théorie des nombres de Bordeaux, 27 no. 1 (2015),
p. 289-307, doi: 10.5802/jtnb.902

Generalization of this paper ("Second Bordeaux paper generalized") pdf

Number of Solutions to $a^x + b^y = c^z$, *
Publ. Math. Debrecen*, Vol 88 (2016), pp. 131-138 .

2018 Publ. Math. Debrecen: Two terms with known prime divisors adding to a power. (Publ. Math. Debrecen, Vol. 93, issue 3-4, 2018, pp. 457-473)

2019 Survey on the Terai-Jesmanowicz conjectures: A Survey on the Ternary Purely Exponential Diophantine Equation $a^x + b^y = c^z$, by Maohua Le, Reese Scott, Robert Styer, Surveys in Mathematics and its Applications, Vol 14, 2019, 109-140.

2022 with Maohua Le, On a conjecture concerning the number of solutions to $a^x+b^y=c^z$, Bulletin of the Australian Math Soc., online 2022.

2013 Note: Comments on the equation **
$\pm r a^x \pm s b^y = c$**

Draft Paper (Extension of Earlier Papers) (Short Version referenced in other papers) The equation $\vert p^x \pm q^y \vert = c$ in nonnegative $x$, $y$

Draft (updated) The equation $\vert p^x \pm q^y \vert = c$ in nonnegative $x$, $y$.

Elementary Treatment of $p^a \pm p^b + 1 = x^2$

Misc including improved bound for Reese's 1993 JNT paper

**Comments and computer calculations for the 2004 JNT paper
"On $p^x - q^y = c$ and related
three term exponential Diophantine equations
with prime bases." Journal of Number Theory, Vol.
105, April 2004, pp. 212-234 **

Here is a slightly improved version of "On $p^x - q^y = c$ and related three term exponential Diophantine equations with prime bases" written with Reese Scott and Robert Styer: pdf. The official version was published in the Journal of Number Theory, Vol. 105, April 2004, pp. 212-234. The version here contains a shorter argument in the third paragraph following equation numbers (30a) and (30b), a minor correction at the end of the proof of Theorem 4, a revised theorem 5, a new version of Lemma 1 allowing a shorter proof of Case 1 of Theorem 8, and some changes in the Observations at the end of the paper.

This paper contains three references to computer calculations:

The calculations that conclude the proof of Theorem 2, and
those that are part
of the discussion about generalizing Theorem 2 at the end of Section 2, claim
that $p^x - q^y = c$
has no double solutions for p, q < 68200 are contained in
this Maple program.

At the end of the paper, we extend the results on $p^x \pm q^y \pm 2^z = 0$ to verify all possible solutions for pairs of odd primes ( p,q) less than 1000. Here is the Maple program and here is an Excel spreadsheet with the 481 solutions, including the five listed in equation (42) of this paper.

The Corollary to Theorem 2 required that the prime p be a Wieferich prime, from which the lower bound of 10^15 arose. Recently, Reese figured out how to use continued fractions to extend the lower bound on $p$ in this Corollary. The lower bound on p is now 10^452, from the least size of a continued fraction in this Maple program (txt version).

**Comments and computer calculations for the 2006 JNT paper "On the generalized Pillai equation
$\pm
a^x \pm
b^y
= c$ " Journal of Number
Theory, 118 (2006), pp. 236-265.**

"On the generalized Pillai equation $\pm a^x \pm b^y = c$" was published in the Journal of Number Theory, 118 (2006), pp. 236-265.

Concerning the last sentence at the bottom of page 246 of the published version of this paper, please note that the case $a^x = 9$ is handled by Theorem 7 of [24]. Also, on page 258, note that the final line ignores the case $a=3$ which is handled in the next paragraph (page 259). Here is a revised version incorporating these corrections and further improvements: pdf. In addition to the corrections mentioned above, Observation 1 has been corrected to include the restriction a>3. We have also significantly simplified Lemmas 3, 4, and 5, and made a minor change following equation (41) in the proof of Theorem 4.

Here is the relevant Maple file (txt
version) that give all double solutions
for a,b<10000 and powers up to 10^20 (we extended this:
here is the Maple file for a,b < 25000, and
here is the file up to 53000). In Theorem 1 of this paper we give all cases of (a,b,c)
which give three or more solutions (x,y) to the equation
$\pm
a^x \pm
b^y
= c$ equation **(1) **where the \pm signs are
independent.

The question remains: for which $(a,b,c)$ does (1) have exactly two solutions $(x,y)$? There are three infinite families of such $(a,b,c)$ discussed in the paper in the Comments following Lemmas 2, 6, and 7. There are also two trivial families of such (a,b,c): the first of these consists of those cases in which a^(x_1) = b^(y_2) and a^(x_2) = b^(y_1); the second consists of those cases in which a=b=2 and x_i = y_j for some 1<= i <= 2 and 1<= j <= 2. If all these infinite families are excluded from consideration, we find at least 138 anomalous cases of (a,b,c) giving exactly two solutions to (1). If we consider $(b, a, c)$ the same as $(a,b,c)$ and disregard duplications due to $a$ or $b$ being a perfect power, then there are still 58 such anomalous cases. These are the only anomalous solutions with terms less than $10^{20}$ when $a , b < 53000$. Noting that, from any $(a,b,c)$ for which (1) has two solutions $(x_1, y_1)$ and $(x_2, y_2)$ such that $x_2 = 2 x_1$, we can derive a new $(a,b,c)$ by using $a^{2 x_1} \pm a^{x_1} = (a^{x_1} \pm 1)^2 \mp (a^{x_1} \pm 1)$, and disregarding rearrangements of terms, we can reduce the number of anomalous cases from 58 to 14. These 14 are:

13 - 3 = 13^3 - 3^7 = 10

91 - 2 = 91^2 - 2^13 = 89

2 + 5 = 2^5 - 5^2 = 7

15 - 6 = 15^2 - 6^3 = 9

5 + 279 = 5^7 - 279^2 = 284

4930 - 30 = 4930^2 - 30^5 = 4900

6^4 - 3^4 = 6^5 - 3^8 = 1215

2^2 + 6^2 = 2^8 - 6^3 = 40

15^2 + 5^2 = 15^3 - 5^5 = 250

11 - 2^2 = 2^7 - 11^2 = 7

40^2 - 2^6 = 2^16 - 40^3 = 1536

5 - 3 = 3^3 - 5^2 = 2

6^2 - 3^2 = 3^5 - 6^3 = 27

21^2 - 98 = 98^2 - 21^3 = 343

All known cases of (a,b,c) giving more than one solution to (1) can be derived either from these 14 cases (which each give exactly two solutions), or from one of the cases listed in Theorem 1 (which give more than two solutions), or from one of the infinite families discussed in the Remark following Theorem 1. The first seven of these 14 equations correspond to the last seven equations in (1.2) of Bennett's paper referenced as [Be] in the paper.

In the proof of Theorem 3 of this same paper, we use a Maple program (txt version) to verify that there are no triple solutions with a,b < 24333. The bounds used are justified by several lemmas which revise and sharpen results in Section 4 of Bennett [Be]. Bennett's Theorem 1.1, which says that the equation $a^x - b^y = c$ has at most two solutions in positive integers (x,y) for a,b >= 2, is proven in Section 3 of [Be] using inequalities derived in that section. His inequalities do not handle (1), hence the need to derive new results paralleling those derived in Section 4 of [Be]. The results in Section 4 of [Be] are used by Bennett not for his Theorem 1.1 but rather for his Theorem 1.3 which deals with the case c >= b^( 2 a^2 log a ) : he shows that a^x - b^y = c has at most one solution for such c. Using our sharpened versions of the results of Section 4 in [Be], we can improve 2 a^2 log a to 2 a log a; however, for values of a < 22, some computation is necessary, so we have not included this result in our paper. (The coefficient 2 can also be reduced.) When a is prime, Bennett obtains the same result for c >= b^a (the specific case (a,b,c) = (2,3,13) is an exception which is not mentioned but it is clear from the context that it is excluded).

Another Maple program (txt version) uses the class number conditions from Reese's 1993 paper to verify the claim in Theorem 6 for p = 1 mod 16 and p<10^8. To facilitate this search, we make use of two congruence conditions: k = 3 or 7 mod 16, and k not congruent 1 mod 3. The first of these congruences follows from the fact that 2^y + 2^(t+1) is divisible by ( p^( (k-1)/2 ) - 1 )/ (p-1) congruent to (k-1)/2 mod 8, so that, since 2^y + 2^(t+1) cannot be divisible by any primes 5 or 7 mod 8, we must have k = 3 or 7 mod 16. The second congruence follows from the fact that if k = 1 mod 3, then 7 divides p^( k-1 ) - 1 which in turn divides 2^y + 2^(t+1).

This note treats the generalization of Theorem 6 when a is composite.

Finally, in the proofs of Lemma 9 and Lemma 11, a bootstrapping program is mentioned. Here is a Maple program to do this, in fact, it covers a,b < 5000, whereas for this article we only use the b=2 and b=3 cases.

**Computer Calculations for
"The generalized Pillai equation $ \pm r
a^x \pm s b^y = c$," Journal of Number Theory, 131 (2011) no. 6,
1037�1047. **

Here is a slightly updated version of the paper "The generalized Pillai equation $\pm r a^x \pm s b^y = c$" (pdf), the official version is Journal of Number Theory, vol. 131, June 2011. This revised version allows the exponents to be zero in Theorem 3.

The paper refers to several computer calculations. The first is a simple-minded search for triple solutions for 1< a, b <,=100 and 1 <= r, s <= 1000, with (ra, sb)=1. Here is the Maple program (pdf, mw) The second is a program that uses simple bootstrapping on each set (r, a, s, b) with 1 < a, b <= 15, 1 <= r, s <= 100, with (ra, sb)=1, to eliminate almost all possible triple solutions to the title equation; the first (pdf, mw) does the bulk of the bootstrapping, and the second finishes the cases that have solutions (pdf, mw).

**Comments on the JTNB paper "The
number of solutions to the generalized Pillai equation $\pm r a^x \pm s b^y = c$" Journal de th�orie des nombres de Bordeaux,
25 no.
1 (2013),
p. 179-210
**

Here is a reformatted version of that paper (pdf). It shows the title equation has four or more solutions (x,y) in only a finite number of cases, except for possible examples within the bounds: a, b, r, s, x, y < 2*10^15. The next paper (Handling a large bound for a problem on the generalized Pillai equation $\pm r a^x \pm s b^y = c$) eliminates these remaining possibilities, proving the following:

Any set of solutions $(a,b,c,r,s; x_1, y_1, x_2, y_2, \dots, x_N, y_N)$ to
$\pm r a^x \pm s b^y = c$ with $N > 3$ must be in the same family as a subset
(or an associate of a subset) of one of the following:

(3,2,1,1,2; 0,0,1,0,1,1,2,2)

(3,2,5,1,2; 0,1,1,0,1,2,2,1,3,4)

(3,2,7,1,2; 0,2,2,0,1,1,2,3)

(5,2,3,1,2; 0,0,0,1,1,0,1,2,3,6)

(5,3,2,1,1; 0,0,0,1,1,1,2,3)

(7,2,5,3,2; 0,0,0,2,1,3,3,9)

(6,2,8,1,7; 0,0,1,1,2,2,3,5)

(2,2,3,1,1; 0,1,0,2,1,0,2,0)

(2,2,4,3,1; 0,0,1,1,2,3,2,4)

**Computations for**** "Handling a large bound for a problem on
the generalized Pillai equation $ \pm r
a^x \pm s b^y = c$" (Integers
Journal 2014, A49)
**

The slightly revised paper (pdf). It refers to several sets of Maple computer calculations.

Maple bootstrapping module (pdf, txt) , LLL basis reduction bound module (pdf, mw, txt)

Lemma 6 (Eqn 3, small values) (pdf, mw)

Lemma 9 (sigma bounds) (pdf, mw)

Lemma 12 (Eqn 4 small values) (pdf, mw)

Lemma 13 (Eqn 4, special instances of (10))

d=1, b<=600000
(pdf,
mw)

the y4 calculations (pdf,
mw)

Lemma 14 (Eqn 4, large values) case with 2 <= x3-x2 < y3-y1 <=4, and case with x3-x2=1 and y3-y1 = 3 or 4 (pdf, mw)

Lemma 15 (Eqn 5, small solns) (pdf, mw)

Looking for three solution instances for Equation (3) for min(a,b) <=
2000, y_3 <= 15.
(pdf,
mw)

Looking for three solution instances for Equation (4) for min(a,b) <= 2000, y_3 <= 15.
(pdf,
mw)

Looking for three solution instances for Equation (4) for min(a,b) <= 100000, y_3 <=
7.
(pdf,
mw)

We also searched for anomalous cases similar to the b=1477 case,
showing there are no other three solution instances for Equation (4) with min(a,b) < 1000000 with y1=1 and y3=4.
(pdf,
mw)

**Computations for "Bennett's Pillai theorem with
fractional bases and negative exponents allowed"
Reese Scott; Robert
Styer
Bennett’s Pillai theorem with fractional bases and negative exponents allowed
Journal de théorie des nombres de Bordeaux, 27 no. 1 (2015),
p. 289-307, doi: 10.5802/jtnb.902
**

This paper (pdf) shows that the equation $a^x - b^y = c$ has at most two solutions in integers $x$ and $y$ when $a$, $b$ and $c$ are rational, except for listed exceptions. Updated version.

The introductory section discusses how elliptic curves sometimes lead to infinite families of two solutions. Here is the main summary: p1; and more supporting calculations: p2, p3, p4.

The end of Chapter 2 refers to some continued fraction calculations (pdf, mw).

We have a generalization of this paper ("Second Bordeaux paper generalized") pdf.

Here is a somewhat shorter streamlined revision of the published paper (pdf).

For relatively prime integers $a$ and $b$ both greater than one and odd integer $c$, there are at most two solutions in positive integers $(x,y,z)$ to the equation $a^x + b^y = c^z$. The most recent published bound, also independent of a, b, c, was a bound of at most 2^36 solutions, announced by Hirata-Kohno using a result of Beukers and Schlickewei. For odd $c$, Le gave a bound of $2^(\omega+1)$ where $\omega$ is the number of distinct prime factors of $c$.

There are an infinite number of
(a,b,c) giving exactly two solutions. We give the following conjecture,
which allows $c$ even as well as $c$ odd:

Conjecture:

For integers $a$,
$b$, and $c$ all greater than one with gcd(a,b) = 1, there is at most one
solution in positive integers (x,y,z) to the title equation except for the
following (a,b,c) or (b,a,c): (5, 2, 3), (7, 2, 3), (3, 2, 11), (3, 2,
35), (3, 2, 259), (3, 4, 259), (3, 16, 259), (5, 2, 133), (3, 10, 13),
(89, 2, 91), (91, 2, 8283), (3, 5, 2), (3,13,2), (3, 13, 4), (3, 13, 16), (3,
13, 2200), and $(2^{n} - 1, 2, 2^{n} + 1)$ for any positive integer $n \ge 2$.

The result that there are at most two solutions to the title equation in the special case in which either $x$ or $y$ is a fixed constant (usually 1) has been obtained by Bennett using lower bounds on linear forms in logarithms when $\gcd(a,b)=1$ and using elementary methods when $\gcd(a,b)>1$. The present paper provides an elementary proof of Bennett's result for the case $c$ odd.

**"Two
terms with known prime divisors adding to a power" Publ. Math. Debrecen, Vol.
93, issue 3-4, 2018, pp. 457-473** (2018)

This paper bounds the number of solutions to $X + Y = c^z$ where $XY$ is
divisible by precisely a given set of primes. Here is a
revised
pdf
of an
extended version of the paper with some arguments streamlined and two appendices added.

We have some calculations of the number of solutions for $n>3$:
relevant examples culled from the
outstanding Matschke database (see
https://www.math.u-bordeaux.fr/~bmatschke/data/ for his data sets).

We also have further calculations of cases with a large number of solutions for certain subsets of
primes:

{2, 3, 5, 7} pdf

{2, 3, 5, 13} pdf

{2, 3, 5, 7, 11} pdf

{2, 3, 5, 7, 13} pdf

{2, 3, 5, 7, 11, 13}
pdf (note the case with six solutions)

{2, 3, 5, 7, 11, 17}
pdf

{2, 3, 5, 7, 13,
17} pdf

{2, 3, 5, 7, 11, 13, 17}
pdf

{2, 3, 5, 7, 11,
13, 17, 19} pdf

Case of $n=2$ bases
pdf

A
presentation on this paper from the Joint Math Meetings January 2022.

A presentation for
MathFest 2018 focusing on Section 6 which discusses the $n=2$ case.

**The Terai-Jesmanowicz conjectures: "A
Survey on the Ternary Purely Exponential Diophantine Equation $a^x + b^y =
c^z$" by Maohua Le et al. Surveys in Mathematics and its Applications,
Vol. 14 (2019), 109-140.
**

Maohua Le invited us to assist in preparing this comprehensive survey of current progress on this equation and on the Jesmanowicz and Terai-Jesmanowicz conjectures. The survey especially summarizes work by Chinese authors that might not otherwise be accessible to non-Chinese speaking mathematicians. We hope this survey will greatly facilitate work on these problems. http://www.utgjiu.ro/math/sma/v14/v14.html (a pdf of the draft version)

**Comments
concerning "On a conjecture concerning the number of solutions to $a^x+b^y=c^z$"
, Bulletin of the Australian Mathematical Society**

In this paper (pdf), we deal with a conjecture from our 2016 paper "On the number of solutions to $a^x + b^y = c^z$. Maohua Le sent us a detailed draft dealing with our conjecture when the bases a,b,c are distinct primes. We added some ideas, in particular, adding a final section where we outline some ideas on proving the conjecture for a range of prime bases ($a, b < 10^9$, or $c<10^{18}$.)

Let $N(a,b,c)$ be the number of solutions $(x,y,z)$ to $a^x + b^y = c^z$ for a given $(a,b,c)$. Here is our conjecture when limited to prime bases:

Conjecture for prime bases:

For $a$, $b$, and $c$ distinct primes
with $a<b$, we have $N(a,b,c) \le 1$, except for

(i) $N(2,3,5) = 2$,
$(x,y,z) = (1,1,1)$ and $(4,2,2)$.

(ii) $N(2, 3, 11)=2$,
$(x,y,z)=(1,2,1)$ and $(3,1,1)$.

(iii) $N(2, 5, 3)=2$, $(x,y,z)=(1,2,3)$
and $(2,1,2)$.

(iv) $N(2, 7, 3)=2$, $(x,y,z)=(1,1,2)$ and $(5,2,4)$.

(v) $N(3,5,2)=3$, $(x,y,z)=(1,1,3)$, $(1,3,7)$, and $(3,1,5)$.

(vi)
$N(3,13,2)=2$, $(x,y,z)=(1,1,4)$ and $(5,1,8)$.

This paper lists a large number of restrictions on possible prime triples (a,b,c). Using these restrictions, we show that when a,b,c are distinct primes with $a < b < 10^9$ or $c < 10^{18}$, we have $N(a,b,c) \le 1$, except for the given $(a,b,c)$ listed in the conjecture. Here is a brief outline of the algorithm used and here is a zip file of the Maple and Sage worksheets used to eliminate $b<10^9$.

This paper summarizes some comments used in a previously listed paper: (pdf)

**Draft paper "The equation
$\vert p^x \pm q^y \vert = c$ in nonnegative $x$, $y$.** "

This paper proves more results on this equation, extending the results of the 2004 and 2006 papers above, for instance, by allowing the exponents to be zero. (pdf)

The main result of the paper deals with primes p and q, but numerical evidence indicates that the result would probably be the same if one allowed composite p and q: as noted above, there are no further solutions for a, b < 53000 with a^x and b^y < 10^20 ( here is the file done previously that does not include two potential exponents equaling zero, here is a supplementary file covering the cases of zero exponents.)

Exploring the equation when the exponents are allowed to be zero: (pdf)

**Draft paper "Elementary Treatment of $p^a \pm p^b + 1 = x^2$" **

Reese Scott simplified work of Szalay and of Luca: here is a draft of his results on "Elementary treatment of $p^a \pm p^b + 1 = x^2$." (pdf)

**Draft paper "Comments on Le's results on the equation $a^x + b^y = c^z$"**

Reese Scott has also noted some easy improvements to results of Maohua Le in his 1999 Proc. Japan Acad. paper. (pdf)

**Comments on Theorem 2 of [Sc] (Reese Scott's
1993 paper)**

Theorem 2 of [Sc] is derived by elementary methods. Using the recent well-known results of [BHV], we can attain an improved version of this theorem, given as Theorem 2A below. For Theorem 2A, we use the notation of Theorem 2 of [Sc], along with the following additional notation:

Let U be the set of all q_i - ( P / q_i) (this is a Legendre symbol), 1 <= i <= n.

Let L be the least common multiple of all the members of U. (Note that L is the final factor in (13) of [Sc].)

Theorem 2A:If (P, Q, r) is not equal to (7, 15, 2), there exists at least one t in U such that, in (13) of Theorem 2 of [Sc], L can be replaced by t. Further, if L > 1, P>2, P is a not a prime congruent to 3 mod 4, and (P, Q, r) not equal (15, 77, 2) or (13, 570, 7), then there exists at least one t in U such that L in (13) can be replaced by t/2, except possibly when (P - 1) congruent (L - 2) congruent 0 mod 4, in which case L can be replaced either by some t/2 or by 2.

(If we simply wish to have an upper bound on x, we take the maximum t; if we wish to obtain a short list of all possible x, we must consider each t.)

[BHV] Y. Bilu, G. Hanrot, P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, With an appendix by M. Mignotte,

J. Reine Angew. Math.,539, (2001), 75--122.

[Sc] R. Scott, On the Equations $p^x-b^y = c$ and $a^x+b^y=c^z$,Journal of Number Theory,44, no. 2 (1993), 153-165.

**Misc info for Reese**

Reese's first 1993 JNT paper

Some Maple calculations for Reese, 19 Dec 2006 (link)

(here are some more 2 July 2007)

Maple imaginary quadratic class number list up to 10000

S versus m calculations Maple file

((p-1)/2)! mod p = +1 or -1 text file

Sketch of Theorem 6 for $a$ composite Revised 5 Jan 2008

West Coast Number Theory 2007 question (newest version)

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