Regarding the Islamic mathematician and astronomer Omar Khayyam, known (among other things) for his accurate calculation of the year, quoted from https://www.famousscientists.org/omar-khayyam/
Khayyam found that 1,029,983 days made 2,820 years. This gives a tropical year length of 365.2422 days to seven significant figures.
I haven't been able to find out how he did such a thing. Clearly he did not wait 2820 years and count the days. So what was the method?
Khayyam simply did not achieve this precision. It is an urban myth, typically lacking any proper sources and references. Only occasionaly one finds mentions such as
"the Persian calendar using the 2820-year arithmetic algorithm, as suggested by Ahmad Birashk and others (and mistakenly attributed to Omar Khayyam)".. [1]
In a paper at the arxiv the author states that
The length of the year in the Iranian calendar, as conceived by Khayyam et al., is 365.2424.. days (Youschkevitch & Rosenfeld 1973), a logical consequence of the intercalation system: 365 + 8 / 33 = 365.2424.
Section 7 of the paper is "The putative 2820-year cycle" where he elaborates:
recently an extremely precise value for the duration of the year (365.24219858156 days) has been attributed to Khayyam (O’Connor & Robertson 1999), although it is supported by no historical source, as far as we know. We think that this may be a spurious value resulting from the erroneous 2820-year cycle suggested recently.
In an other paper at the arxiv there is just a hint at the issue mentioning The researches of two modern Iranian scholars. These two scholars are Zabīh Behrūz and Ahmad Birashk :
They have accepted a system of intercalation based on a 2820 years principal cycle with its own 128 years sub- cycles, 29 or 33 years sub-sub-cycles, “tetraennial”s, and “ pentaennial”s.[2]
So it appears that they relied on modern 19th c. data to obtain a cycle with an integer number of days and next proceeded to implement an intercalation scheme that matches older practices. (For the record Meton solved an analogous problem for the luni-solar calendar). There is no good explanation how to explain the 4 years added at the end of a series of 22 repetions of 128 years sub cycles, except that the number 2820 is known in advance. The precision needed to arrive at it does not seem possible without sophisticated time keeping devices and observational instruments as the equinox should be clocked with an error less than a half of second. (Averaging data could produce by chance the decimals needed.)
He used the same principle as Hipparchus and all other astronomers, including those more ancient than Hipparchus.
The idea is the following: if you can determine some fixed moment in the year (like a solstice or an equinox) with error E, then by doing this N years apart, you divide the number of days between two successive events on N, and your error will be E/N. So larger the N is better your accuracy is. For example, if you know the only the number of days from one equinox to another 1000 years later, your can determine the length of a year with accuracy 1/1000 of a day.
We don't know exactly what ancient observations Hipparchus used (but it is almost certain that he had access to some ancient Babylonian observations few hundred years before him. Hipparchus own observations are preserved by Ptolemy.
But Omar Khayyam had a much longer period at his disposal, at least from Hipparchus to his own time. (And more, since Ptolemy also preserved more ancient observations). Accordingly his accuracy was higher. Speaking of the techniques of fixing the exact moment of a solstice or equinox, there was no much improvement from the times of Hipparchus to the time of Khayyam. So using my notation, their E was of the same order of magnitude, while Khayyam's N was much larger.
You can obtain even better accuracy by computing the time from the earliest observation known to you to an equinox which you can observe yourself.
