+/- r a^x +/- s b^y = c goal: to find all cases of three solutions.Eqn (21) case A Lemma 14 calculationFor 5 <= b <=100000 (actually need (8 * 10^14)^(1/3)), 2 <= x3-x2 < y3-y1 <= 4. In all cases assume a>b and assume y3-y1 > x3-x2.We consider sets of three solutionsNotation: 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 Eqn (2) yields these for (i,j) = (1,2) and (2,3) and (1,3):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
We then bound y3 using the sigma bound of Lemma 11: b^y3 | b^\134sigma_b(a) * (x4-x3) < b^\134sigma_b(a) * Z < b^\134sigma_b(a) * 8 * 10^14. so y3 < sigma_b(a) + log(8*10^14)/log(b).26 Oct 20104 Nov 201017 Jan 20115 April 201113-19 Apr 201126-29 Apr 20112-31 May 20111-5, 8 July 201122-28 Oct 2011LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic=LUkld2l0aEc2IjYjSSpudW10aGVvcnlHRiQ=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=QyQ+SS1maW5kcG93ZXJkaXZHNiJmKjYkSSNtMUdGJUkiY0dGJTYjSSRhbnNHRiVGJUYlQyY+SSNtMkdGJTkkPjgkIiIhQCQyIiIiRi4/KEYlRjVGNUYlLy1JJG1vZEdGJTYkRi45JUYyQyQ+Ri4qJkYuRjVGOyEiIj5GMSwmRjFGNUY1RjVGMUYlNiNGLkYlRj8=QyQ+SSlwb3dlcm1vZEc2ImYqNiVJJWJhc2VHRiVJJHBvd0dGJUknbW9kbnVtR0YlRiVGJUYlLUkkbW9kR0YlNiQtSSMmXkdGJTYkOSQ5JTkmRiVGJUYlISIiQyQ+SStmaW5kbWF4ZGl2RzYiZio2JkkibUdGJUkibkdGJUkkcG0xR0YlSSJjR0YlNiRJImlHRiVJJGFuc0dGJUYlRiVDJT44JSIiIUAkLy1JJG1vZEdGJTYkLCYtSSlwb3dlcm1vZEdGJTYlOSQ5JTknIiIiOSZGP0Y+RjJDJD8oOCRGP0Y/LUklY2VpbEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLUkmZXZhbGZHRkc2IyooRj1GPy1JJGxvZ0dGRjYjRjxGPy1GTzYjRj4hIiIvLUY2NiQsJi1GOjYlRjxGPSlGPkZDRj9GQEY/RlpGMkYlPkYxLCZGQ0Y/RlNGP0YxRiVGJUYlRlM=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=We first deal with the case when x_3-x_2 =2, y3-y1=3. We only need to check b < (8 * 10^14)^(1/3) < 10^5. Note the a=18, b=7 and a=19, b=7, instances below are really from the (10) case by setting b = 7^3 to make it fit the form given in (10). Note the a=20, b=7 instance below is really from the (10) case by reversing the a and b and in (10) letting a=7, b=20^2. 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For the y3-y1=4 case, we only need to check b < (8 * 10^14)^(1/4) < 10^4.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZecC1JI21vR0YkNi1RIn5GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGQy1JI21pR0YkNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEqJmNvbG9uZXE7RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjI3Nzc3NzhlbUYnL0ZFRlNGKy1GRzYlUSV0aW1lRidGSkZNLUkobWZlbmNlZEdGJDYkLUYjNiQtRkc2I1EhRidGL0YvLUYsNi1RIjtGJ0YvRjIvRjZGTEY3RjlGO0Y9Rj9GQUZULUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRkMvJSZkZXB0aEdGY28vJSpsaW5lYnJlYWtHUShuZXdsaW5lRidGKy1GRzYlUSZ5ZGlmZkYnRkpGTUZPRistSSNtbkdGJDYkUSI0RidGL0ZqbkYrRitGK0Zeb0Zeb0YrLUYsNi9RJGZvckYnLyUlYm9sZEdGTC9GMFElYm9sZEYnLyUrZm9udHdlaWdodEdGaHBGMkY1RjdGOUY7Rj1GP0ZBRkRGKy1GRzYlUSJiRidGSkZNRistRiw2L1ElZnJvbUYnRmVwRmdwRmlwRjJGNUY3RjlGO0Y9Rj9GQUZERistRl9wNiRRIjVGJ0YvRistRiw2L1EjdG9GJ0ZlcEZncEZpcEYyRjVGN0Y5RjtGPUY/RkFGREYrRistRl9wNiRRJjEwMDAwRidGL0YrLUYsNi9RI2RvRidGZXBGZ3BGaXBGMkY1RjdGOUY7Rj1GP0ZBRkRGXm9GK0YrRmJwRistRkc2JVEmeGRpZmZGJ0ZKRk1GK0ZecUYrLUZfcDYkUSIyRidGL0YrRmRxRitGW3AtRiw2LVEoJm1pbnVzO0YnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yMjIyMjIyZW1GJy9GRUZnci1GX3A2JFEiMUYnRi9GK0ZqcS1GX282JkZhb0Zkb0Zmby9GaW9RJWF1dG9GJ0Zeb0YrLUZHNiVRKW1haW5wcm9jRidGSkZNLUZZNiQtRiM2JEZbcUYvRi9Gam5GXm9GKy1GLDYvUSRlbmRGJ0ZlcEZncEZpcEYyRjVGN0Y5RjtGPUY/RkFGREYrRmpxRmpuRl5vRitGZ3NGK0ZqcUZqbkZeb0YrRltxRmpuRitGVUZYRmNyRkZGam4vJStleGVjdXRhYmxlR0Y0Ri8=Now we need to bootstrap to show that the b=1477 cases do not have fourth solutions. We first assume the fourth solution has the same +/- signs as the third, then assume the signs are opposite. Begin with a=56744, b=1477. 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 shows x4 exceeds 8*10^14 so no fourth solution 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this shows y4 exceeds 8*10^14 so no fourth solution possibleNow we need to deal with the plus sign cases. Note that the order of a mod s*b^4 is odd, so there is no solution to a^x + 1 = 0 mod s b^4. Next, we consider the second case, with a=56745, b=1477. 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this shows x4 exceeds 8*10^14 so no fourth solution 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 shows y4 exceeds the bound, so no fourth solution is possible.Now we need to consider the case of plus signs.Note r=739 divides b^(y4-y3) + 1 which cannot have a solution mod 739. 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JSFHWe have now finished the x3-x2=2 or 3 cases. We now begin the x3-x2=1 case, y1 < y3-y1 = 3 or 4 cases. We changed notation in the paper: now it is 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Then Eqn (2) yields these for (i,j) = (1,2) and (2,3) and (1,3):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where xdiff = 1. 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Again we have the bound b < * 8*10^14 )^(1/ydiff) so for ydiff=3 we have b<10^5 and for ydiff=4 we have b<10^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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFH