Fermat’s Last Theorem

In 1637 Pierre de Fermat wrote in the margin of Diophantus’s Arithmetica the statement that would puzzle some of the world’s greatest mathematicians for over three centuries:

It is impossible to write a cube as a sum of two cubes, a fourth power as a sum of two fourth powers, and, in general, any power beyond the second as a sum of two similar powers. For this, I have discovered a truly wondrous proof, but the margin is too small to contain it.

Fermat, celebrated for making such declarations with little confirmation, kept mathematicians scratching their heads and squaring their roots trying to discover proofs for his statements. By the 19^th century, all of Fermat’s theories had been resolved except the one above, a statement that became know as Fermat’s Last Theorem.

We will never know whether Fermat had actually discovered a correct proof of his theorem, but we do know that Andrew Wiles of Princeton University produced a 130-page proof in 1994, 357 years after Fermat wrote his tantalizing marginal note.

Although Fermat’s Last Theorem has not yet been used for practical purposes, many new ideas and numerous practical technological advances developed in solving the problem. Sometimes what we learn along the way to a destination becomes more important than reaching the end of our journey.  

                                                                  Revised from The Heart of Mathematics by Edward Burger and Michael Starbird

Trintellix, A Novel Antidepressant

Trintellix (vortioxetine) formerly named Brintellix is a multimodal antidepressant recently approved by the FDA for the treatment of major depressive disorder that has a number of pharmacologic effects marked by the following characteristics:

• Increases the release of serotonin, norepinephrine, dopamine, glutamate, acetylcholine, and histamine while reducing the release of GABA
• Blocks serotonin reuptake
• Binds to G protein-linked receptors
• Binds to ion channel-linked receptors
• Full agonist at the 5-HT_1A receptor provides an anxiolytic and antidepressant effect while diminishing sexual side effects
•  Partial agonist at the 5-HT_1B receptor associated with low weight gain
•  Antagonist at 5-HT_1D receptor associated with low weight gain and improved cognition
•  Antagonist at the 5-HT₃ receptor contributes to antidepressant effects and improves cognitive function by enhancing noradrenergic, acetylcholinergic, and glutamatergic activity and reduces nausea and vomiting caused by serotonin reuptake inhibition
• Antagonist at the 5-HT₇ receptor improves cognitive function, learning, and memory and contributes to antidepressant effects

These multimodal neurotransmitter effects contribute to usefulness in major depressive disorder, generalized anxiety disorder, cognitive symptoms associated with depression, and geriatric depression.

Clinical studies suggest that vortioxetine has a good safety and tolerability profile: Sexual dysfunction was low; vortioxetine had no significant effect on weight; and adverse events such as insomnia, fatigue, sedation, and somnolence were also low. Nausea, vomiting, and headache were the most common side effects.

Onset of action may be faster than most other antidepressants, with a clinical response in two weeks perhaps.

The recommended starting dose of Trintellix is 10 mg administered orally once daily without regard to meals. After one week dosage should then be increased to 20 mg daily, because higher doses demonstrated better treatment effects. A decrease down to 5 mg daily may be considered for patients who fail to tolerate higher doses. The long half-life (66-hours) of Trintellix allows the drug to be discontinued abruptly although prudence dictates tapering down to 10 mg daily for one week.

Treatment of Bipolar Depression

On average people with Bipolar I Disorder spend three times more in a depressive episode than in a manic or hypomanic state; those with Bipolar II Disorder spend 40 times more in a depressive episode than in a manic or hypomanic state. According to DSM-5 a bipolar depressive episode and a unipolar depressive episode have the same diagnostic criterea marked by five of the following nine symptoms for two weeks:

• Depressed mood
• Loss of pleasure
• Weight or appetite change
• Change in sleep
• Psychomotor agitation or retardation
• Fatigue
• Feelings of worthlessness or guilt
• Indecisiveness or trouble concentrating
• Suicidal ideation

Bipolar I patients have had at least one manic episode lasting at least one week during which time the person feels euphoric and overly optimistic or is extremely irritable marked by three of the following symptoms:

• Unrealistic, grandiose beliefs about one’s abilities or powers
• Rapid speech that makes it difficult for others to keep up
• Acting recklessly without thinking about the consequences
• Racing thoughts, jumping quickly from one idea to the next
• Distractibility marked by poor concentration and attention
• Impulsiveness, poor judgment, agitation or excessive goal pursuit
• Sleeping very little but without loss of energy

Bipolar II patients have had at least one hypomanic episode that differs from mania by intensity and duration. A hypomanic episode is only required to persist for four days instead of seven. Those in a hypomanic state can make bad decisions that harm relationships, careers, and reputations, but they are able to perform without losing touch with reality.

Three medications have been approved by the FDA for the treatment of acute bipolar depressive episodes:

     1. Seroquel (quetiapine)

     2. Symbyax (a combination of fluoxetine and olanzapine)

     3. Latuda (lurasidone)

Seroquel can contribute to significant weight gain and metabolic syndrome as can Symbyax. Both Latuda and Symbyax are expensive.

Lithium, listed as first-line treatment for acute bipolar depression in the APA practice guidelines, works rapidly, is inexpensive, and substantially decreases suicide risk. Because lithium has been a generic medication for decades there is little incentive for pharmaceutical companies to seek FDA approval.

Lithium, like ECT, has a negative connotation for some people so preparing patients for this gold standard treatment requires education. For example the clinician can say: “Lithium is a natural element. You’ll find it on the periodic chart. Dosing is more exacting than with most other medications because we can measure blood levels regularly. Like all medications lithium has side effects but we will routinely monitor this side effect potential with a series of laboratory studies.”

Once the educational groundwork has been laid the clinician can prescribe 300 mg immediate-release lithium at night. (Data shows that immediate-release once daily is better for kidney function.) After five days the dose can be increased to 600 mg. Following another five days the dose can be increased to 900 mg. Baseline labs and a lithium level can be drawn within a week of starting lithium.