The lower bound of 6 was later improved to 11 by Geoff Exoo in 2003, and to 13 by Jerome Barkley in 2008. F7(12)  =  F(F(F(F(F(F(F(12))))))),{F^{7}(12)\;=\;F\big(F(F(F(F(F(F(12))))))\big)},F7(12)=F(F(F(F(F(F(F(12))))))), Net current asset value per share (NCAVPS) is a measure created by Benjamin Graham as one means of gauging the attractiveness of a stock. {\displaystyle 3\uparrow \uparrow \uparrow 3\ =\ 3\uparrow \uparrow (3\uparrow \uparrow 3)}.3↑↑↑3 = 3↑↑(3↑↑3). Take a chance and explore the math of unpredictability. 3↑↑↑3 = 3↑↑(3↑↑3). Log in here. Graham's Number was invented by Ronald Graham in 1971 as the upper bound for a problem used as a mathematical proof. It is named after mathematician Ronald Graham who used the number as a simplified explanation of the upper bounds of the problem he was working on in conversations with popular science writer Martin Gardner. Therefore, the first level of up-arrow notation is just exponentiation, and we can write 3↑43 \uparrow 43↑4 as 34.3^4.34. Color each of the edges of this graph either red or blue. Strengthen your algebra skills by exploring factorials, exponents, and the unknown. N′  =  2  ↑↑↑  6. It is calculated by dividing the total net asset value by the number of shares outstanding. However, he found that Graham’s number was much easier to explain, and so he used it instead when talking to Gardner. You still won't be able to reach Grahams number. It's defined recursively, with the base case of repeated multiplication. Even power towers of the form abc⋅⋅⋅{\displaystyle \scriptstyle a^{b^{c^{\cdot ^{\cdot ^{\cdot }}}}}}abc⋅⋅⋅ Value investors like Warren Buffett select undervalued stocks trading at less than their intrinsic book value that have long-term potential. It is used as a general test when trying to identify stocks that are currently selling for a good price. ) operation reduces to a power tower ( 03222348723967018485186439059104575627262464195387.3881448314065252616878509555264605107117200099709291249544378887496062882911725063001303622934916080254594614945788714278323508292421020918258967535604308699380168924988926809951016905591995119502788717830837018340236474548882222161573228010132974509273445945043433009010969280253527518332898844615089404248265018193851562535796399618993967905496638003222348723967018485186439059104575627262464195387. Essentially, this second method of calculation is equivalent to the first, wherein EPS = net income/shares outstanding, and book value is another term for shareholders’ equity. But one number that I haven’t written about is the Graham Number, another conservative way to look at stock values. ↑↑ ↑↑{\displaystyle \scriptstyle \uparrow \uparrow }↑↑ It is so large that the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume which equals to about 4.2217×10−105 m34.2217\times 10^{-105}\text{ m}^{3}4.2217×10−105 m3. {\begin{matrix}3^{3^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{3}}}}}}\end{matrix}}\right\}\left. Graham was working on a combinatorics question and found a proof involving an extremely large number. The Graham number is the upper bound of the price range that a defensive investor should pay for the stock. It is named after mathematician Ronald Graham who used the number as a simplified explanation of the upper bounds of the problem he was working on in conversations with popular science writer Martin Gardner. {\begin{matrix}3^{3^{\cdot ^{\cdot ^{\cdot ^{3}}}}}\end{matrix}}\right\}\dots \left. For example, the Graham Number - the price calculation for Defensive quality stocks - is calculated as: Services and other asset-light companies were common in Graham's … To convey the difficulty of appreciating the enormous size of Graham's number, it may be helpful to express, in terms of exponentiation alone, just the first term g1g_1g1​ of the rapidly growing 64-term sequence. What the Price-To-Book Ratio (P/B Ratio) Tells You? Graham's number, G,G,G, is much larger than N:N:N: f64(4),{f^{64}(4)},f64(4), where f(n)  =  3↑n3. Benjamin Graham one proposed a quick back-of-the-envelope intrinsic value formula for investors to determine if their stocks were at least somewhat rationally priced. ) alone, ↑↑ ↑↑{\displaystyle \scriptstyle \uparrow \uparrow }↑↑ Guided training for mathematical problem solving at the level of the AMC 10 and 12. The price-to-book ratio (P/B ratio) evaluates a firm's market value relative to its book value. g1=33⋅⋅⋅⋅3}33⋅⋅⋅3}…333}3,where the number of towers is33⋅⋅⋅3}333}3,{\displaystyle g_{1}=\left. The number gained a degree of popular attention when Martin Gardner described it in the "Mathematical Games" section of Scientific American in November 1977, writing that Graham had recently established, in an unpublished proof, "a bound so vast that it holds the record for the largest number ever used in a serious mathematical proof.". What is the smallest value of nnn for which every such coloring contains at least one single-colored complete subgraph on four coplanar vertices? where a superscript on an up-arrow indicates how many arrows there are. The formula is as follows: 22.5 × (earnings per share) × (book value per share)\sqrt{22.5\ \times\ \text{(earnings per share)}\ \times\ \text{(book value per share)}}22.5 × (earnings per share) × (book value per share)​. The term is also sometimes referred to as Benjamin Graham’s number. Anyway other specific integers known to be far larger than Graham's number have since appeared in many serious mathematical proofs, for example, in connection with Harvey Friedman's various finite forms of Kruskal's theorem.