The North, which says it needs nuclear weapons to protect itself against the threat of a USA invasion, responded on Friday with a rare personal rebuke from Kim, who called Trump "mentally deranged" and threatened the "highest level of hard-line countermeasure in history".
The U.S.is putting on display its military might for North Korea. Former President Barack Obama made similar comments during his administration when he said the U.S.
All North Korea's nuclear tests registered as earthquakes of magnitude 4.3 or above.
The back-and-forth messages between North Korea and the United States intensified this weekend as the US military sent bombers and fighter escorts to the farthest point north of the border between North and South Korea.
U.S. bombers accompanied by fighter jets flew off the east coast of North Korea Saturday in a show of force created to project American military power in the face of Pyongyang's weapons programs, the Pentagon said.
American bomber and fighter jets flew along North Korea's eastern coastline Saturday in a "show of force" that was closer to the rogue nation's border than any other mission this century, the Pentagon reported.
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It does raise many questions, including how would the North undertake such a nuclear test, what risks might it pose to Japan and how would the US respond?
The patrols came after officials and experts said a small natural disaster near North Korea's nuclear test site on Saturday was probably not man-made, easing fears Pyongyang had exploded another nuclear bomb just weeks after its last one.
Kim in turn called Trump a "frightened dog" and a "mentally deranged USA dotard".
Mr Trump on Friday had renewed his rhetorical offensive against Kim.
Kim Jong Un of North Korea, who is obviously a madman who doesn't mind starving or killing his people, will be tested like never before!
On Thursday Trump announced new USA sanctions that he said allows the targeting of companies and institutions that finance and facilitate trade with North Korea.
Pyongyang conducted its sixth and largest nuclear test on September 3 and has launched dozens of missiles this year as it accelerates a program aimed at enabling it to target the United States with a nuclear-tipped missile.
In a separate development, China moved to limit the North Korea's oil supply and stop buying textiles from the country, in line with the latest United Nations sanctions.