On Polygonal Square Triangular Numbers

A pentagonal square triangular number is a number which is a pentagonal number P5(ℓ), a square y2 and a triangular number P3(m) at the same time. It would be well known for the specialists that there exists no pentagonal square triangular number except for P3(1) = 12 = P5(1) = 1. But we don’t know any simple reference of the proof of this fact in print. The object of this note is to provide a such reference. Here we shall present three independent proofs of this fact one of which was already referred in the net article [24]

Exploring the Base of the Volcano: A Case Study of an Active Stratovolcano, Mt. Zao, NE Japan

It is very important to explore the base of large volcanoes because older volcanoes with distinct petrological characteristics are sometimes hidden behind them. Such older volcanoes provide keys to investigate the change of magma genesis and tectonic setting during geological time. We newly found an older volcano in southern part of Zao volcano, located in Japan. We have investigated in detail the eruptive products outcropping in its southern part and found that some eruptive rocks with peculiar features form a new stratovolcano which is different from Zao volcano. We call this newly found volcano, the Hiyamizuyama volcano. We have performed K-Ar dating on the representative rocks, obtaining an old age of approximately 1.45 My. The rocks are calcalkaline andesites to dacites, having distinct chemical compositional features with respect to any other calcalkaline rock of the stages 2–6 of Zao volcano. Megacrystals and plutonic intrusions represent a distinct character of the eruptive rocks of the Hiyamizuyama volcano. The finding of this older volcano is also important in order to consider the long-term temporal variation of volcanism and magmatism in the northeastern sector of Japan