Arithmagicians are often overlooked when it comes to being labeled as wizards. Obsessed with calculating the true limitations of the arcane and the world it influences, these wizards are always seen pouring over numbers. Things often believed to be chance are typically understood by Arithmagicians to be an enigmatic equations hidden by a level of arcane influence. Everything and anything can be a number.

Arithmagicians are known to be the most pedantic of wizards. Due to the nature of their study, everything must be precisely calculated and recorded unless the world devolve into chaos. What brings an Arithmagician greater joy than solving an equation is coming up with a theory for a new one.

Logbook

2nd-level Arithmagician feature

Mathematics and incredible theory learned by probing the depths of the arcane have become your weapons. You maintain a log with all your knowledge.

While you are holding the logbook, it grants you the following benefits:

• You can use the book as a spellcasting focus for your wizard spells.
• When you cast a wizard spell with a spell slot, you may use an algorithm gained by another feature to change the results of the spell.
• When you cast a wizard spell as a ritual, you may roll a d10 and reduce the casting time by that amount after adding 10 minutes to it. 

Arcane Algorithms

2nd-level Arirthmagician feature

As a bonus action, you can magically imbue a spell you cast with an algorithm. You may choose one effect for the algorithm from the following:

• Roll a d4. The spell cast must target that many additional creatures(yourself included) or the spell and spell slot is lost. The spell must be incapable of already targeting more than one target. 
• Roll a d6. Gain advantage on an attack roll on even number. Give a target disadvantage on saving throw on an odd number.

Fickle Figures

6th-level Arithmagician feature

Beginning at 6th level, as a reaction, you may roll a d8 and choose to make the result impossible or doubled. If impossible, the number is treated as a 1 whenever you roll it on any roll of the die. If more dangerous, the number is considered 2x the value rolled. (i.e an 8 is counted as a 16). Only one number may be used at a time.

Quick Math

10th-level Arithmagican feature

Before you roll an attack or saving throw, you may choose to call out a number. Roll an additional d12. If the number is within a certain range of the called number, the attack roll is counted as critical or the saving throw is a success. The range is determined by the d12. The called number can be above or below the attack roll or saving throw as long as it is within the range determined. 

This can be done equal to INT modifier per long rest.

Prime Power

14th-level Arithmagican feature

Your dedication to numbers has uncovered that certain numbers have unique potential. The prime numbers, as you label them, are capable of creating extraordinary effects when applied to or removed from the creation of spells. 

Whenever you cast a spell, you may roll a d100. You must designate prime or not prime before the roll is made. Apply an effect based on the number rolled and if it was designated.

Prime number designated and rolled: The power surges through you duplicating the spell instantly. Cast the spell again. You may choose new targets. 

Prime number designated and not rolled: You remove the priming material of the spell and cast the spell without material components. 

Nonprime number designated and rolled: You achieve the quickest casting of the spell possible. You may cast an additional spell as a bonus action.

Nonprime number designated and not rolled: Your calculations were off, but you are able to recover somewhat. Take force damage equal to half the number rolled and deal the same damage to another target within 60ft. If this were to bring you to 0 hp, it instead brings you to 2 hp.