Apple (NASDAQ: AAPL) shares enjoyed a healthy 2% rally on Tuesday after activist investor Carl Icahn tweeted about his dinner with Tim Cook. Icahn said he aggressively pushed for a $150 billion repurchase program, which would be a significant increase from the current $60 billion authorization.

A $150 billion buyback program would be the size of's entire market cap. Apple has bought back about $18 billion through June, and likely repurchased more shares this quarter. Investors will have to wait until Apple reports earnings later this month to see how much the company repurchased. After last quarter's aggressive repurchase, don't be surprised if Apple bought back another hefty number of shares.

Since Apple is so large, it's not realistic for Icahn to accumulate a 10% stake to enact any possible changes. Icahn has said his position is valued around $2 billion, and a 10% stake would be nearly $45 billion. Still, Icahn could theoretically wage a proxy fight by rallying other shareholders behind him and getting their votes. The investor has said he has no intentions of doing this though, in part because he's quite satisfied with Tim Cook's leadership decisions.

In this segment from today's episode of Tech Teardown, Erin Kennedy discusses Icahn and Apple with Evan Niu, CFA.

Erin Kennedy owns shares of Apple. Evan Niu, CFA, owns shares of Apple. The Motley Fool recommends and Apple. The Motley Fool owns shares of and Apple. Try any of our Foolish newsletter services free for 30 days. We Fools may not all hold the same opinions, but we all believe that considering a diverse range of insights makes us better investors. The Motley Fool has a disclosure policy.