Computer calculations for the 2004 JNT paper on three term exponential Diophantine equations with prime bases
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: tex version, and pdf version. 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 revision 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 not 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 for the special case 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
Reese and I have also worked on another paper which has been published in the
Journal of Number Theory, 118 (2006), pp. 236-265:
"On the generalized Pillai equation
$\pm
a^x \pm
b^y
= c$."
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: TeX version and pdf version. 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.
Elementary Treatment of $p^a \pm p^b + 1 = x^2$
Reese Scott has been extending 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)
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
Some Maple calculations, 19 Dec 2006 (Click on this link)
(here are some new ones 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 Sketch 22 Dec 2007
Computer results 31
Dec 2007
Sketch of
Theorem 6 for $a$ composite Revised
yet again 5 Jan 2008
West Coast Number Theory question (newer version)
Return to Robert Styer Home Page
06/29/2009: