School of Arithmancy

2nd level: Arithmancy Savant
Through years of studying the underlying formulas and equations that are used to channel specific combinations of magical energy, you have gained the ability to see significantly more connections between spells and their structure than other wizards.

When copying a spell into your spellbook, the Gold Pieces and time needed to copy it is reduced by half if your spellbook already contains a spell of the same level and school. When copying a spell into your spellbook that meets this requirement, you can further reduce the amount of time needed to only 25% of the time needed without this feature by expending 2 spell slots of a level equal to the spell. This additional optional feature will not further reduce the Gold needed.

6th level: Mana factorization
Your studies into the numerical aspects of spells have left you with insight into manipulating the raw energy used to cast them.

When you cast a spell that can be cast at any level higher than its base level, you may expend a charge of Arcane Recovery as part of casting the spell.

If you do, the level of spell slot you expend to cast the spell is the base level of the spell, while you still gain the benefits of casting the spell at the higher level. This feature can only be used to gain the effect of casting a spell at a certain level if you have a spell slot of that level.

10th level: Prime Factorization Formula
Your extended study of Mana Factorization and further understanding of spell structure and rules has increased your ability to manipulate magical energies.

You gain one extra Charge of the Arcane Recovery feature, but it must still follow all rules of that feature when used.

In addition, when using an Arcane Recovery Charge to use the Mana factorization feature, you immediately also gain the benefit of using the Arcane Recovery Feature, but with a maximum total of spell slot levels that you can recover being 2 lower than if you had used the Arcane Recovery normally.

14th level: Energy Density Alteration Formula
You have spent much of your life solving many equations with complex solutions, comparing the overlaps and interactions between different types of energy and their densities within spells.

Whenever you complete a long rest, you may reduce the number of spells you can prepare for that day by 2 to gain an additional spell slot of any level 5 or lower. You can use this feature to gain a maximum of 3 additional spell slots. Any unused additional spell slots gained by this feature are lost when you start your next long rest.

Arithmancy Savant

When copying a spell into your spellbook, the Gold Pieces and time needed to copy it is reduced by half if your spellbook already contains a spell of the same level and school. When copying a spell into your spellbook that meets this requirement, you can further reduce the amount of time needed to only 25% of the time needed without this feature by expensing at least 2 spell slots of a level equal to the spell. This additional optional feature will not further reduce the Gold needed.

Mana Factorization

When you cast a spell that can be cast at any level higher than its base level, you may expend a charge of Arcane Recovery as a free action as part of casting the spell.

If you do, the level of spell slot you expend to cast the spell is equal to the base level of the spell, while you gain the effects of having cast it at a higher level of your choice, with the maximum level that you can choose the spell to take effect at being the highest level spell slot you have.

 

Example: By expending one Arcane recovery charge, you can gain the effect of casting a Fireball at 5th level while only using a 3rd level spell slot, but only if you have access to 5th level spell slots.

Prime Factorization Formula

You gain one additional Charge of the Arcane Recovery feature, but it must still follow all rules of that feature when used.

In addition, when expending an Arcane Recovery Charge to use the Mana factorization feature, you immediately also gain the benefit of using the Arcane Recovery Feature, but with the maximum total level of spell slot(s) that you can recover being reduced by 2 compared to if you had used the Arcane Recovery normally.

Energy Density Alteration Formulas

Whenever you complete a long rest, you may reduce the number of spells you can prepare for that day by 2 to gain an additional spell slot of any level 5 or lower. You can use this feature to gain a maximum of 3 additional spell slots. Any unused additional spell slots gained by this feature are lost when you start your next long rest.