The number on the screen just before Max faints and is revived by his neighbours shows 218 digits, not 216.
The note suddenly appears on the Go board between shots when Max enters Sal's living room.
When Max sees ABR stock price in the newspaper in the subway, it is 6 1/2 but net change from previous day is +2, a large increase, as opposed to a big drop. It was predicted by Euclid that ABR will fall to 6 1/2 after trading above 40 for years, so net change should have been -35 or something.
When Max is examining the sticky ant remains with his fingers, the substance switches from being on his middle finger to being on his index finger in the close-up.
Max suggests that the Kabbalists have already tried all possible 216-digit numbers. That is 10 to the power of 215 multiplied by 9 [9x(10^215)] - if they had been doing a quadrillion a day since the destruction of the Temple, they would barely have begun.
Several times in the movie the numbers following a decimal place are called "integers". Integers are always whole numbers and never fractions or decimals.
Max' ticker displays some stock prices in thirds, but real stock prices only used halves, quarters, eights, sixteenths, and so on.
In the opening credits, Pi is written out as a long string of digits, but it begins to diverge from the actual value of Pi at the ninth decimal place, showing 3.14159265263 where the correct digits are 3.14159265358.
Even given the eons of time needed to write down and intone all the 216-digit numbers, there isn't room. The surface area of the earth is approximately 510 million square kilometers, or 510 * 10^6 sq. km, which, given 10^12 square millimeters to each square km, is 510 * 10^18 square mm of surface area. If you divide 10^216 numbers by the total surface area of the earth, you would still need to write over 10^190 numbers in each square millimeter of space. Since a square millimeter can contain less than 10^15 atoms of hydrogen, the impossibility of this task is obvious.
When Robeson is discussing Archimedes' Principle with Max, he states that Archimedes wanted to assess density by measuring weight and volume.
Nowadays, density is (usually, not always!) defined as mass, rather than weight, per volume so as to make it independent of the strength of the prevailing gravitational field.
The concept of gravity, however, did not arise for many hundreds of years after Archimedes. It is, therefore, not a goof to say that Archimedes used the crown's weight to calculate density rather than its mass.
Max shaves his head and draws a square around the bulge on the side of his head. Later on, when Max is chasing a photographer from the subway, there is no square on his head. This could be due to time shifts in the story, and hallucinations had by Max. Making the actual linear time of events impossible to tell for sure.
When Max is in the coffee shop, the cream container he opens has a McDonald's logo. It's a cheesy enough place that they were probably ripped off.
In different takes the "Wall-Street" numbers on the display panel are the same, the numbers never really change and it is more or less always the same stocks, no matter what time it is.
Shadow of camera operator visible looming over Max as he thrashes on the floor of his bathroom.
Max mentions that the Golden Mean is represented by the Greek letter theta. In fact it is denoted using the Greek letter Tau or, more commonly Phi, whereas phi is used to denote 1/Phi. Phi is for Phidias, a 5th Century BC Greek sculptor who employed the ratio.
When Lenny demonstrates basic examples of Hebrew numeric coincidences, he gets the numbers right but mistranslates some of the Hebrew. "Kedem" does not mean "Garden of Eden" but is a biblical word means "East" (as in "East of Eden"). Also, "Etz Hakhaim" is not the tree of knowledge but rather the other tree in the garden, the tree of life.
Max indicates that he medicates himself with "Promazine 'HCI'." The proper name of the drug is "Promazine HCl," where the HCl stands for hydrogen chloride. The error likely occurred when either Max or a filmmaker mistook a lowercase "L" for a capital "I" on a bottle or prescription.
Max writes the golden ratio as (a/b) = (a/a+b). It should be a/b = (a+b)/a, if, by definition, a>b.
The scene where Max is rebuilding his computer after stomping on it shows him soldering components onto various circuit boards. Both the soldering iron and the solder he is using is of the wrong type for that type of work. The tip on the iron is way too large which would not only prevent him from performing accurate work, it would also transfer too much heat to the integrated circuits, risking serious damage. No one with even a basic knowledge of soldering would use an iron like that for that purpose.