Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. This paper contains the proof that every positive composite integer n strictly...Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. This paper contains the proof that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes, which implicitly proves Goldbach’s Conjecture for 2n as well.展开更多
The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact...The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes. The present paper contains explicit additional and complementary details of the proof, insisting on the existence and the number of Goldbach’s representations of even positive integers as sums of pairs of primes.展开更多
The p27Kip1 is a cell cycle repressor protein that regulates primarily the cell cycle transition from G1 to S phase and hence the DNA replication is in the S phase and cell division in the M phase. Expression of p27Ki...The p27Kip1 is a cell cycle repressor protein that regulates primarily the cell cycle transition from G1 to S phase and hence the DNA replication is in the S phase and cell division in the M phase. Expression of p27Kip1 protein has dual roles for both cancer prevention and promotion. For example, numerous nutritional and chemopreventive anti-cancer agents specifically increase the expression of p27Kip1 protein without directly affecting the expression of any other cell cycle regulatory proteins. On the other hand, pro-cancer agents (like glucose, insulin and other growth factors frequently seen in obesity and/or diabetes) specifically decrease the expression of p27Kip1 protein without directly affecting the expression of any other cell cycle regulatory proteins. Unlike expression of any other cell cycle regulatory proteins, expression of p27Kip1 protein is very unusual. The mRNA of p27Kip1 has a very long and unusual 5’-untranslated region (from -575 to -1 in human). It appears that the 5’-untranslated region of p27Kip1 mRNA forms two alternative secondary structures. One increases the expression of p27Kip1 protein when anti-cancer agents are added and another decrease the expression of p27K1p1 when pro-cancer agents are added. For this short concept proposal, Dr. Albert Einstein’s “visualized thought experiments (German: Gedanken experiment)” were used as a fundamental tool for understanding how either anti- or pro-cancer agents bring the primary structure of the 5’-untranslated region of p27Kip1 mRNA into two alternative secondary structures, thereby either increasing or decreasing, respectively, the translation initiation of p27Kip1 protein.展开更多
文摘Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. This paper contains the proof that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes, which implicitly proves Goldbach’s Conjecture for 2n as well.
文摘The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes. The present paper contains explicit additional and complementary details of the proof, insisting on the existence and the number of Goldbach’s representations of even positive integers as sums of pairs of primes.
文摘The p27Kip1 is a cell cycle repressor protein that regulates primarily the cell cycle transition from G1 to S phase and hence the DNA replication is in the S phase and cell division in the M phase. Expression of p27Kip1 protein has dual roles for both cancer prevention and promotion. For example, numerous nutritional and chemopreventive anti-cancer agents specifically increase the expression of p27Kip1 protein without directly affecting the expression of any other cell cycle regulatory proteins. On the other hand, pro-cancer agents (like glucose, insulin and other growth factors frequently seen in obesity and/or diabetes) specifically decrease the expression of p27Kip1 protein without directly affecting the expression of any other cell cycle regulatory proteins. Unlike expression of any other cell cycle regulatory proteins, expression of p27Kip1 protein is very unusual. The mRNA of p27Kip1 has a very long and unusual 5’-untranslated region (from -575 to -1 in human). It appears that the 5’-untranslated region of p27Kip1 mRNA forms two alternative secondary structures. One increases the expression of p27Kip1 protein when anti-cancer agents are added and another decrease the expression of p27K1p1 when pro-cancer agents are added. For this short concept proposal, Dr. Albert Einstein’s “visualized thought experiments (German: Gedanken experiment)” were used as a fundamental tool for understanding how either anti- or pro-cancer agents bring the primary structure of the 5’-untranslated region of p27Kip1 mRNA into two alternative secondary structures, thereby either increasing or decreasing, respectively, the translation initiation of p27Kip1 protein.
