A ‘mathematical prophecy’ in the Alhambra?

I was walking quietly through the gardens of the Alhambra (that is to say, quietly), trying to escape from that crowd that made it difficult to imagine what the daily life of emirs, sultans, viziers, princesses, eunuchs, concubines, in short, the entire Andalusian court was like. of the moment. It was the third time I was in Granada, and the one that I have been able to enjoy the most and lose myself in its wonderful corners. Among the many thoughts that those places inspire was the privilege that it must have been to have lived there, to see sunrise and sunset from within, as did the romantic Washington Irving, no wonder that in such an environment he wrote such a beautiful collection of stories and legends. . Unfortunately, Irving was not a mathematician, nor has it been recorded that he handled them. Only in this way is it explained that he will not realize the enormous potential that each and every one of its nooks has. However, in the Andalusian court there were scientists, geometricians, engineers and astrologers (what are we going to do; it was a time when the terms astronomy and astrology were not differentiated). What would each of the members of that court think under a five-month long siege and lack of supplies? Boabdil, like all kings, emperors, rulers, etc., of all times, places and conditions (I refer to History, let no one be angry, but that’s the way it is; there are few exceptions), I would be plotting how to save the hide, having private conversations with his opponents to resolve the conflict in the best possible way. But mathematicians, those of the past and those of now, would be giving the head to see how to help in such a critical situation, although surely they would pay little attention to them in the end (as History also teaches us). And astrologers, also to their own thing, to mount spells, curses and all kinds of tricks that they don’t even believe (well, at that time, maybe they did). This preamble comes to the fact that on my visit I realized, in one of the many closed corridors behind a fence that we find throughout the different rooms of the complex (surely fast escape routes, as in practically all castles and fortresses) , I noticed, almost at ground level and scratched on the mortar (which, letting my imagination run wild, suggests that they were done quickly and running), of somewhat enigmatic numerical characters. Fortunately, current technology (the mobile camera) allowed me to take a photograph that I later analyzed more calmly. After several days of thinking about it (one is a mathematician, not a paleographer), I concluded that they indicated, expressed in our current terminology: Below, almost indistinguishable, other figures appear. The clearest is 777. At first, I thought it was a simple count, because those four numbers were placed on top of each other, as we do today. In addition, at first the exponent went unnoticed, so it seems simply addition and subtraction, perhaps a game or something similar since the result is zero: 1492 – 1898 – 1936 + 2342. You know that I do not believe in anything that It has to do with esotericism, or other nonsense (for me; I respect those who believe in them, but they are nonsense). The first number, 1492, clearly indicates the year in which the Catholic Monarchs took possession of the place. And here the unsettling thing begins: if that number indicates a year, there is no reason to think that the others are not as well. And call me crazy, but, with little history that one knows, 1492 is the year of the surrender of the city, 1898 and 1936 are dates of unfortunate memory for the Spaniards (two wars; the first with the total loss of the colonies and the civil war). Thinking about it more carefully, it is also curious that they are written following the Gregorian calendar, so that the conquerors were aware of a message. Maybe a curse? What will 2342 mean? It seems like the end of something (it’s the last date), maybe the end of the world? Although, they are all dates related to our country: the end of Spain? Anyway, nonsense as I said before. But let’s get to the math part. It happens that, giving n natural values ​​(remember, 1, 2, 3, …), the expression is always divisible by 777 (the number that appears separately), and I suppose you know the symbolism that cultures like the Muslim assigned to 7 Also three times: “Allah has created 7 heavens, 7 earths and 7 days.” As soon as they have handled powers of integers (and those are four-digit integers), they will have realized that there comes a time when their value grows so much that it quickly becomes unmanageable. How did they show, for any natural n, that this expression is divisible by 777? Perhaps by induction? I don’t get anything like that (or I don’t see it). Are there more numbers that divide that expression, for any n? Someone may be thinking, “What nonsense! Surely 777 is a common factor of the four numbers. Well don’t look, it’s not so trivial. I leave you the factorizations of those numbers so that you do not have to waste time on that: I hope that, throughout this week, some of you will be able to answer any of the questions (of course those of a mathematical nature; Whether 2342 has any special meaning does not matter to us a bit: overall, none of us will make it to that year). As a curiosity, so that you can realize how the numbers potentially grow, if you try to substitute n for 2022, the year we start, you will see that it is a 6814 digit number. Perhaps it would be a good idea to write some ‘Mathematical Tales of the Alhambra’. For what it’s worth, I have already left you one of them, because all this is a story inventing to remember in the new year that mathematics is a certainty. Happy New Year! Alfonso Jesús Poblacion Sáez is a professor at the University of Valladolid and a member of the Disclosure Commission of the Royal Spanish Mathematical Society (RSME). The ABCdario de las Matemáticas is a section that arises from collaboration with the RSME Disclosure Commission. .